L_{p}-properties of second order elliptic differential operators

DISSERTATION

zur Erlangung des akademischen Grades

Doctor rerum naturalium (Dr. rer. nat.)

vorgelegt

der Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden

von

Dipl.-Math. Hendrik Vogt

geboren am 8. 6. 1971 in Oldenburg (Oldb)

Gutachter: Prof. Dr. J. Voigt Prof. Dr. V. Liskevich Prof. Dr. K.-Th. Sturm

Eingereicht am: 15. 2. 2001 Tag der Disputation: 29. 6. 2001

Contents

Introduction 3

1 C_{0}-semigroups and sesquilinear forms 7 1.1 C_{0}-semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Analytic semigroups and sectorial forms . . . . . . . . . . . . . . 9 1.3 Perturbation of positive C_{0}-semigroups . . . . . . . . . . . . . . . 11 1.4 The first Beurling-Deny criterion . . . . . . . . . . . . . . . . . . 15 1.5 Dirichlet forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Analyticity and L_{p}-spectral independence 28 2.1 Framework and main results . . . . . . . . . . . . . . . . . . . . . 28 2.2 Comments and Examples . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Technique of weighted estimates . . . . . . . . . . . . . . . . . . . 41 2.4 Extrapolation and analyticity . . . . . . . . . . . . . . . . . . . . 46 2.5 L_{p}-spectral independence . . . . . . . . . . . . . . . . . . . . . . . 51

3 Elliptic differential operators 57 3.1 Construction of the semigroup . . . . . . . . . . . . . . . . . . . . 57 3.2 Quasi-contractive C_{0}-semigroups . . . . . . . . . . . . . . . . . . . 67 3.3 Weighted estimates for second order operators . . . . . . . . . . . 73 3.4 Sharpness of the results . . . . . . . . . . . . . . . . . . . . . . . . 84

Bibliography 92

Introduction

The aim of this thesis is to study the L_{p}-theory of (formal) second order elliptic differential operators with singular measurable coefficients. The main tool of our study is the theory of strongly continuous semigroups (C_{0}-semigroups) of bounded linear operators on Banach spaces. It is well-known that, if A_{p} is an operator realisation of a formal differential expression in L_{p} with ρ(A_{p}) ≠ ∅, then the abstract Cauchy problem

u′(t) + A_{p}u(t) = 0 (t > 0), u(0) = f

has a unique solution for all f ∈ D(A_{p}) if and only if −A_{p} is the generator of a C_{0}-semigroup T_{p} on L_{p}. In this case, the solution u is given by u(t) = T_{p}(t)f.

Another motivation to study L_{p}-theory is that it often yields significant information for weak L_{2}-solutions of the corresponding elliptic and parabolic equations, such as integrability or smoothness properties of eigenfunctions and solutions.

We confine ourselves to the case of real-valued coefficients so that the corresponding semigroups will be positive. Our main interest lies in the case of singular coefficients where the first problem one faces is constructing a C_{0}-semigroup on a suitable L_{p}-space, associated with the differential expression.

In the case of a uniformly elliptic principal part of the differential expression, and bounded coefficients in the lower order terms, the associated semigroup exists in the whole L_{p}-scale, which follows, for instance, from classical estimates on the fundamental solution of the corresponding parabolic equation [Aro67]. If the coefficients of the lower order terms are allowed to have strong singularities then a semigroup associated with the differential expression can be constructed in L_{p} for p from a proper subinterval of [1, ∞) only. This phenomenon was first observed in the study of Schrödinger operators with singular negative potentials ([HeSl78], [KPS81]), later also for the operator −Δ + b · ∇ in [KoSe90].

In this thesis we study general second order elliptic expressions in divergence form with both first and zero order perturbations, namely

L := −∇ · (a∇) + b_{1} · ∇ + ∇ · b_{2} + V,

on an open set Ω ⊆ ℝ^{N}, for a wide class of boundary conditions. Generalising results from [BeSe90], [Lis96], we establish a precise condition controlling the interval of those p ∈ [1, ∞) for which L gives rise to a quasi-contractive C_{0}-semi-group T_{p} = (e^{−tA_{p}}; t ⩾ 0) on L_{p}(Ω).

By the Lumer-Phillips theorem, T_{p} is quasi-contractive, i.e., there exists ω ∈ ℝ such that the operators e^{−ωt}T_{p}(t) are contractive, if and only if A_{p} is quasiaccretive. Thus, it is very natural to expect that the condition controlling the interval of quasi-contractivity involves only the expression ⟨Lu, u^{p−1}⟩ occurring in the definition of quasi-accretivity, computed in a suitable sense (see page 59). We have to overcome several technical problems in order to show that this natural condition is indeed sufficient.

In general, the set of all p ∈ [1, ∞) such that L is associated with a (not necessarily quasi-contractive) C_{0}-semigroup on L_{p} is strictly larger. It was already observed in [KPS81] that the Schrödinger semigroup with negative L_{\frac{N}{2} ,weak}-potential can be defined on L_{p} for p from an interval strictly larger than the interval of quasi-contractivity. In [Sem00] this result was extended to uniformly elliptic second order divergence type operators on ℝ^{N} perturbed by a form small potential. Here we show that this behaviour is typical for rather general second order uniformly elliptic operators.

A traditional way of constructing a semigroup associated with the differential expression L is the form method. Since L is given in divergence form, it corresponds to a sesquilinear form in L_{2}. If this form is densely defined, sectorial and closed then it is associated with an m-sectorial operator A in L_{2}, which in turn generates a C_{0}-semigroup (e^{−tA}; t ⩾ 0) on L_{2} ([Kat80; Thms. VI.2.1, IX.1.24]). If ||e^{−tA}↾_{L_{2}∩L_{p}} ||_{L_{p}→L_{p}} ⩽ Ce^{ωt} for some p ∈ [1, ∞) then (e^{−tA}↾_{L_{2}∩L_{p}}; t ⩾ 0) extends to a semigroup on L_{p}. For p > 1, this semigroup is always strongly continuous, whereas for p = 1 this is the case if, e.g., the semigroup is positive (see [Voi92]).

The above approach was used for constructing semigroups which act in L_{p} for all p ∈ [1, ∞) (see [Dav89; Chapter 1] and the references there), or only for p from some subinterval of [1, ∞) containing 2 (see, e.g., [BeSe90], [Lis96]). However, we do not assume that the form corresponding to L is sectorial, not even that it is bounded from the left, so the traditional approach is not applicable. In the case b_{2} = 0, V = 0, non-sectorial forms were studied in [KoSe90], [Lis96] by approximating the coefficient b_{1} in such a way that the approximating forms became sectorial.

Here we develop a new approach to the construction of positive C_{0}-semi-groups associated with sesquilinear forms. It includes cases of forms that can be associated with a C_{0}-semigroup on L_{2} under assumptions when all known representation theorems break down.

Our approach is based upon approximations by sectorial forms, however, not related to approximations of the coefficients of the first order terms. In contrast, we approximate the potential: we introduce a positive potential U which ‘absorbs’ all the singularities of the lower order terms of L in the sense that, being added to the corresponding form, it makes the sum sectorial. Under certain conditions, this gives rise to a positive C_{0}-semigroup on L_{p}, as described above. Finally, making use of the perturbation theory for positive semigroups developed in [Voi86], [Voi88], we subtract the potential U again. It is crucial for this construction that the resulting semigroup turns out to be independent of the particular choice of U.

In the context of Schrödinger operators with magnetic fields, and dominated semigroups with singular complex potentials, a similar approximation idea was used in [PeSe81] and in [LiMa97]—however, not in order to construct semigroups but to study properties of semigroups constructed in a different way.

The L_{p}-properties of the semigroups we study here include analyticity with pindependent sector, and p-independence of the spectra of the generators. Assume that the differential expression L is associated with a C_{0}-semigroup on L_{p}, for p from some subinterval of [1, ∞). If one of the semigroups is analytic then by the Stein interpolation theorem one can show that, for p from the interior of this interval, the semigroup on L_{p} is analytic; the resulting angle of analyticity, however, tends to zero as p approaches the endpoints of the interval.

E. M. Ouhabaz [Ouh95] was the first to establish analyticity of angle \frac{π}{2} in L_{p}(ℝ^{N}), p ∈ [1, ∞), for symmetric semigroups satisfying Gaussian upper bounds. E. B. Davies [Dav95a] extended this result to the more general setting of metric spaces with polynomial volume growth. For symmetric semigroups acting on L_{p} for p from a subinterval of [1, ∞) only, analyticity of angle \frac{π}{2} was first shown in [Sch96], under the assumption of certain weighted estimates. Here we prove an analogous result for general uniformly elliptic second order operators.

Concerning the problem of L_{p}-spectral independence, let us mention that this is not a general property of second order elliptic operators (see, e.g., [HeVo86]). However, as first dicovered by R. Hempel and J. Voigt, it is generic for Schrödinger type operators. Here we present rather general conditions on the coefficients of second order elliptic operators under which p-independence of the spectrum holds.

In order to treat the three problems described above, namely extension of the semigroup to L_{p}, analyticity, and L_{p}-spectral independence, we first provide a general setting in which these problems can be studied, and we formulate proper sufficient conditions in terms of weighted norm estimates. Motivated by the paper [Dav95a] mentioned above, we study semigroups on metric spaces with exponentially bounded volume growth, not just on open subsets of ℝ^{N}. This enables us to unify and generalise numerous previous results concerning the three problems under consideration—see the discussion in Section 2.2.

As a specific application of the abstract results we obtain that the Schrödinger semigroup on a Riemannian manifold with Ricci curvature bounded below is analytic of angle \frac{π}{2} on L_{p}, for p from a certain subinterval of [1, ∞) (p ∈ [1, ∞) if the negative part of the potential is in the Kato class). This result seems to be new even for positive potentials—in this case the semigroup operators act as contractions on all L_{p}-spaces.

Many of the known results are proved under the assumption that the semigroup acts on all L_{p}-spaces and has an integral kernel satisfying a Gaussian upper bound. Weighted norm estimates, i.e. estimates on the norm of the semigroup operators as operators between weighted L_{p}-spaces, were first used in [ScVo94] to establish L_{p}-spectral independence for Schrödinger operators on ℝ^{N} with form small negative part of the potential. In this case the semigroup acts on L_{p}(ℝ^{N}) only for p from an interval around p = 2.

The proof in [ScVo94] relied on a discrete method where ℝ^{N} is subdivided into congruent cubes. It is clear that this method is essentially restricted to the study of semigroups acting on (subsets of) ℝ^{N} since in the case of a general metric space there is no natural partition into countably many subsets. Here we present a continuous version of the technique of weighted norm estimates which is suitable for the general context.

The structure of the thesis is as follows. To a large extent, Chapter 1 is a collection of known facts about C_{0}-semigroups and their generators (Section 1.1), sectorial forms and the associated analytic semigroups (Section 1.2), Dirichlet forms and the associated sub-Markovian semigroups (Section 1.5), and perturbation of positive semigroups by real-valued potentials (Section 1.3). In Section 1.4, the heart of the chapter, we provide the method that, as described above, is needed in Chapter 3 to construct semigroups on L_{p} associated with (non-sectorial) forms.

Chapter 2 is devoted to the theory of weighted norm estimates for semigroups on metric spaces with exponentially bounded volume growth. In Section 2.1 we present our abstract results on extrapolation, analyticity and L_{p}-spectral independence. In Section 2.2 we give some account to the history of these three problems, and we relate our results to the existing literature. As an application we study perturbation of sub-Markovian semigroups satisfying Gaussian upper bounds (such as the diffusion semigroup on a Riemannian manifold) by potentials. The proofs of the main theorems are given in Sections 2.3-2.5.

In Chapter 3 we apply the abstract theory of Chapter 2 to our main subject, the L_{p}-theory of second order elliptic differential operators. The main results concerning the construction of the semigroup on L_{p}, quasi-contractivity and analyticity of the semigroups, and L_{p}-spectral independence of the generators are formulated in Section 3.1 and proved in the two subsequent sections. In Section 3.4 we study to what extent the assumptions of our theorems are necessary. The main result in this direction is that, for a wide class of coefficients, we can characterise the set of all p for which the differential expression L is associated with a quasi-contractive C_{0}-semigroup on L_{p}.

Acknowledgements. In the first place, I would like to express my gratitude to my supervisor, Prof. Dr. Jürgen Voigt. He always was open for discussing problems, and his way of approaching problems in a structured way had a great influence on me. It was Prof. Dr. Vitali Liskevich who drew my attention to the subjects presented in this thesis. I am grateful to him for inspiring discussions during several visits to Bristol. I thank Amir Manavi and Zeev Sobol for many valuable discussions and remarks. My research was partially supported by the Deutsche Forschungsgemeinschaft. The support is gratefully acknowledged.

Chapter 1

C_{0}-semigroups and sesquilinear forms

In this chapter we provide the functional analytic tools needed in the two subsequent chapters. In Sections 1.1 and 1.2 we collect basic facts about C_{0}-semi-groups, analytic semigroups and sectorial forms, mainly in order to fix our notation but also to recall results frequently used in this thesis. In Section 1.3 we recall J. Voigt’s perturbation theory for positive C_{0}-semigroups. This is fundamental for Section 1.4 where we investigate sesquilinear forms τ in L_{2}(μ) fulfilling the first Beurling-Deny criterion and show how to associate with τ a positive C_{0}-semigroup on L_{p}(μ). Section 1.5 deals with the theory of (non-symmetric) Dirichlet forms, the main example being the form corresponding to a homogeneous second order elliptic differential operator with real coefficients.

1.1 C_{0}-semigroups

In this section we recall some basic definitions and results from the theory of C_{0}- semigroups, the main references being [Dav80] and [Paz83]. Let X be a Banach space over ℂ (throughout this thesis we assume ℂ to be the underlying scalar field). By L(X) we denote the space of all bounded linear operators from X to X.

Definition 1.1. (a) A function T: [0, ∞) → L(X) is called a semigroup on X if T (0) = I and T (s + t) = T (s)T (t) for all s, t ⩾ 0. We say that T is exponentially bounded if ||T (t)|| ⩽ Ce^{ωt} for some C ⩾ 1, ω ∈ ℝ and all t ⩾ 0.

(b) A semigroup T is called strongly continuous or a C_{0}-semigroup if T (t)f → f as t → 0 for all f ∈ X.

(c) The generator of a C_{0}-semigroup T is the operator A in X defined by

D(A) := \{u ∈ X; Au := lim_{t→0} \frac{1}{t}(T (t)u − u) exists\}.

8 CHAPTER 1. C_{0}-SEMIGROUPS AND SESQUILINEAR FORMS

Every C_{0}-semigroup is exponentially bounded. An exponentially bounded semigroup T is strongly continuous if and only if T (t)f → f for all f from some dense subset of X.

The generator of a C_{0}-semigroup T is a closed densely defined operator which determines the semigroup T uniquely. We can therefore write e^{tA} := T (t) for the semigroup operators. From time to time we will loosely speak of a C_{0}-semigroup T (t) = e^{tA} in order to express that T is a C_{0}-semigroup with generator A.

The type of a C_{0}-semigroup T is the infimum of all ω ∈ ℝ for which there exists C ⩾ 1 such that ||T (t)|| ⩽ Ce^{ωt} for all t ⩾ 0. If ω_{0} ∈ [−∞, ∞) is the type and A the generator of T then

λ ∈ ρ(A), (λ − A)^{−1} = ∫_{0}^{∞} e^{−λt}T (t) dt for all λ ∈ ℂ with Re λ > ω_{0},

where the integral is a strong integral.

In the literature, a semigroup T is often denoted by (T (t); t ⩾ 0), (T (t))_{t⩾0} or T (·) in order to indicate that T depends on one parameter. We will mostly use the symbol T only.

Let A be a linear operator in X. The numerical range of A is the set

Θ(A) := {x′(Au); u ∈ D(A), ||u|| = 1, x′ ∈ X′, ||x′|| = 1, x′(u) = 1}.

For X = L_{p}(μ), where (Ω, μ) is a measure space and 1 < p < ∞, we obtain

Θ(A) = {⟨Au, |u|^{p−1} sgn u⟩; u ∈ D(A), ||u||_{p} = 1}.

Here and in the sequel, ⟨f, g⟩ is defined as ∫_{Ω} f(x) · \overline{g}(x) dμ(x) whenever f · \overline{g} ∈ L_{1}(μ), for f, g: Ω → ℂ measurable.

The operator A is called m-accretive if Θ(A) ⊆ {z ∈ ℂ; Re z ⩾ 0} and −1 ∈ ρ(A). It is called quasi-m-accretive if ω + A is m-accretive for some ω ∈ ℝ. A semigroup T is called contractive if ||T (t)|| ⩽ 1 for all t ⩾ 0, and quasicontractive if ||T (t)|| ⩽ e^{ωt} for some ω ∈ ℝ and all t ⩾ 0. The Lumer-Phillips theorem states that −A generates a contractive C_{0}-semigroup if and only if A is m-accretive.

Let E, F be Banach spaces, and assume that there exists a Hausdorff topological vector space G such that E ↪ G, F ↪ G (continuous injections) and E ∩ F is dense in both E and F. Let B_{E} and B_{F} be bounded operators in E and F, respectively. We say that B_{E} and B_{F} are consistent if B_{E}↾_{E∩F} = B_{F} ↾_{E∩F}. Two semigroups T_{E}, T_{F} on E, F, respectively, are called consistent if the operators T_{E}(t) and T_{F} (t) are consistent for all t ⩾ 0.

Let T_{E} be a semigroup on E, D a dense subset of E ∩ F, and assume that T_{E}(t)↾_{D} extends to a bounded operator T_{F} (t) on F, for all t ⩾ 0. Then T_{F} is a semigroup on F, and T_{E} and T_{F} are consistent. In this case we will say that T_{E} extrapolates to the semigroup T_{F} on F.

Later on, we will make use of the following notion.

Definition 1.2. ([Voi86; Def. 1.5]) Let T_{n} (n ∈ ℕ), T be C_{0}-semigroups on X. We say that T_{n} converges strongly to T, in symbols T = s-lim T_{n} or simply T_{n} → T, if T_{n}(t)f → T (t)f as n → ∞, uniformly for t in bounded subsets of [0, ∞), for all f ∈ X.

Let A_{n}, A be the generators of T_{n}, T, respectively. Then T_{n} → T if and only if A_{n} → A in the strong resolvent sense, by the Trotter-Kato-Neveu theorem.

1.2 Analytic semigroups and sectorial forms

For the theory of sectorial forms we refer to [Kat80; Chapter VI], for the connection with analytic semigroups to [Kat80; Sec. IX.1]. For symmetric forms see also [Dav80; Sec. 4.2], for analytic semigroups also [Dav80; Sec. 2.5] and [Paz83; Sec. 2.5].

Definition 1.3. (a) For θ ∈ (0, \frac{π}{2}] let

S_{θ} := {0 ≠ z ∈ ℂ; |arg z| < θ}.

A function T: S_{θ} → L(X) is called exponentially bounded if ||T (z)|| ⩽ Ce^{ω Re z} for some C ⩾ 1, ω ∈ ℝ and all z ∈ S_{θ}.

(b) A C_{0}-semigroup T on X is called analytic if T has an exponentially bounded analytic extension to S_{θ}, for some θ ∈ (0, \frac{π}{2}]. The supremum θ_{0} of such θ is called the angle of analyticity of T. The extension of T to S_{θ_{0}} will again be denoted by T.

Analytic semigroups are usually defined in a different way (see, e.g., [Dav80]), but actually these definitions are equivalent: If a semigroup T has an analytic extension to S_{θ} for some θ ∈ (0, \frac{π}{2}] then T (z + w) = T (z)T (w) for all z, w ∈ S_{θ}, by unique analytic continuation. If T is strongly continuous, and θ < \frac{π}{2}, then the analytic extension to S_{θ} is exponentially bounded if and only if lim_{S_{θ}∋z→0} T (z)f = f for all f ∈ X. For the “only if” part note that {T (t)f; t > 0, f ∈ X} is dense in X and that T is strongly continuous on t + \overline{S_{θ}}, for all t > 0.

A linear operator A in X is called m-sectorial (of angle θ) if Θ(A) ⊆ \overline{S_{θ}} − ω and −1 − ω ∈ ρ(A) for some ω ∈ ℝ, θ ∈ (0, \frac{π}{2}). In particular, if A is m-sectorial then A is quasi-m-accretive. Moreover, −A generates an analytic semigroup of angle at least \frac{π}{2} − θ, with ||e^{−zA}|| ⩽ e^{ω Re z} for all z ∈ S_{\frac{π}{2} −θ}. However, if e^{−tA} is an analytic semigroup then A need not be m-sectorial (the diffusion semigroup on L_{1}(ℝ^{N}) is analytic of angle \frac{π}{2} but its generator is not m-sectorial).

Let H be a Hilbert space. A C_{0}-semigroup T on H is symmetric, i.e. all semigroup operators T (t) are selfadjoint, if and only if the generator of T is selfadjoint. In this case, T is in particular quasi-contractive, and analytic of angle \frac{π}{2}.

A sesquilinear form τ in H is called symmetric if τ(u, v) = \overline{τ(v, u)} for all u, v ∈ D(τ). A symmetric form τ is said to be bounded below if τ ⩾ −ω for some ω ∈ ℝ, i.e.,

τ(u) := τ(u, u) ⩾ −ω||u||^{2} (u ∈ D(τ)).

In this case, τ(u, v) + (ω + 1)(u, v) defines a scalar product on D(τ), and τ is called closed if D(τ) is a Hilbert space with respect to this scalar product. In the following, if we speak of a closed symmetric form τ then this implicitly means that τ is bounded below.

If τ is a symmetric form which is bounded below then we define τ(u) := ∞ for u ∈ H \ D(τ). With this definition, τ is closed if and only if u ↦ τ(u) is a lower semicontinuous function from H to (−∞, ∞] (see [Dav80; Thm. 4.12]). Moreover, if we state an inequality of the type τ(u) ⩽ c then this implicitly expresses that u ∈ D(τ).

Let τ be a sesquilinear form in H. We define the form τ^{∗} in H by τ^{∗}(u, v) := \overline{τ(v, u)} on D(τ^{∗}) := D(τ). Moreover,

Re τ := \frac{τ + τ^{∗}}{2} and Im τ := \frac{τ − τ^{∗}}{2i} ,

where the sum of two forms τ_{1}, τ_{2} is defined by (τ_{1} +τ_{2})(u, v) := τ_{1}(u, v)+τ_{2}(u, v) on D(τ_{1} + τ_{2}) := D(τ_{1}) ∩ D(τ_{2}). Then Re τ and Im τ are symmetric forms, and τ = Re τ + i Im τ, τ^{∗} = Re τ − i Im τ. Note that (Re τ)(u) = Re(τ(u)) for all u ∈ D(τ). (But not (Re τ)(u, v) = Re(τ(u, v)) for all u, v ∈ D(τ)!)

The numerical range of τ is the set

Θ(τ) := {τ(u); u ∈ D(τ), ||u|| = 1}.

The form τ is symmetric if and only if Θ(τ) ⊆ ℝ (recall 𝕂 = ℂ). Moreover, Re τ ⩾ −ω for some ω ∈ ℝ if and only if Θ(τ) ⊆ {z ∈ ℂ; Re z ⩾ −ω}. In this case, τ is said to be bounded from the left.

We say that τ is sectorial if Θ(τ + ω) ⊆ \overline{S_{θ}} for some ω ∈ ℝ, θ ∈ (0, \frac{π}{2}), or equivalently,

|Im τ(u)| ⩽ tan θ(Re τ + ω)(u) (u ∈ D(τ)).

In this case we endow D(τ) with the form norm || · ||_{τ} defined by ||u||_{τ}^{2} := (Re τ + ω + 1)(u). It is easy to see that this definition does not depend on the particular choice of ω, up to equivalence of norms.

We say that a sectorial form τ is closed if the symmetric form Re τ is closed. By Kato’s first representation theorem ([Kat80; Thm. II.2.1]), every densely defined closed sectorial form in H is associated with an m-sectorial operator A in H in the sense that D(A) ⊆ D(τ) and

(Au, v) = τ(u, v) for all u ∈ D(A), v ∈ D(τ).

The operator A is selfadjoint if and only if τ is symmetric. We will shortly write τ ↔ T to indicate that T is the analytic semigroup generated by −A. With this notation, τ^{∗} ↔ T^{∗} := (T (t)^{∗}; t ⩾ 0).

1.3. PERTURBATION OF POSITIVE C_{0}-SEMIGROUPS 11

Let τ be a closed sectorial form, A the associated m-sectorial operator in H. By [Kat80; Cor. 2.3] the numerical range Θ(A) of A is a dense subset of the numerical range Θ(τ) of τ. Therefore, −A generates a contractive C_{0}-semigroup if and only if A is m-accretive, and the latter is true if and only if Re τ ⩾ 0.

A sectorial form τ is called closable if it has a closed extension τ̃ ⊇ τ. The smallest closed extension of τ is denoted by \overline{τ}. The sum of two closable (closed) sectorial forms is again a closable (closed) sectorial form. A subspace D ⊆ D(τ) is called a core for τ if D is dense in D(τ) with respect to the form norm. If τ is closable then this is equivalent to \overline{τ↾_{D}} = \overline{τ}, where we shortly write τ↾_{D} for the restriction of τ to D × D.

Let τ_{1}, τ_{2} be symmetric forms. We write τ_{1} ⩽ τ_{2} if D(τ_{1}) ⊇ D(τ_{2}) and τ_{1}(u) ⩽ τ_{2}(u) for all u ∈ D(τ_{2}). (This is consistent with the definition of τ ⩾ −ω if −ω is interpreted as the form defined by −ω(u, v) with domain H.)

Let H = L_{2}(μ) for some measure space (Ω, μ), and V: Ω → ℝ measurable. Then ∫ V u\overline{v} dμ on the domain Q(V) := \{u ∈ L_{2}(μ); V |u|^{2} ∈ L_{1}(μ)\} defines a symmetric form which we will also denote by V. The form V satisfies V ⩾ −ω for some ω ∈ ℝ if and only if V ⩾ −ω a.e., and V is closed in this case.

We conclude this section by a lemma which is useful for extrapolation of analytic semigroups.

Lemma 1.4. Let (Ω, μ) be a measure space, D a subspace of L_{1}(μ)∩L_{∞}(μ) which is dense in L_{p}(μ) for all 1 ⩽ p < ∞, and norming for L_{1}(μ). Let 1 ⩽ p, q < ∞, S ⊆ ℂ open, F: S → L(L_{q}(μ)) an analytic function. If ||F (·)↾_{D}||_{p→p} is locally bounded then F (·)↾_{D} extends to an analytic function F_{p}: S → L(L_{p}).

Proof. It is clear that F (·)↾_{D} extends to a locally bounded function F_{p}: S → L(L_{p}). The assumption implies that ⟨F_{p}(·)f, g⟩ is analytic for all f, g ∈ D. A slight modification of [Kat80; Thm. III.3.12] shows that F_{p} is analytic. (For the case p = 1 note that D is not necessarily dense in L_{∞}(μ); it suffices that D is a norming subspace for L_{1}(μ).)

This result will be applied in Chapter 3 in the following situation: (Ω_{n}) is an increasing sequence of measurable subsets of Ω such that μ(Ω_{n}) < ∞ and Ω = ⋃_{n} Ω_{n} (in particular, Ω is σ-finite). Then the space of all f ∈ L_{∞}(Ω) for which there exists n ∈ ℕ such that f = 0 a.e. on Ω \ Ω_{n} is a suitable choice for D.

1.3 Perturbation of positive C_{0}-semigroups by real-valued potentials

In this section we give a short introduction to J. Voigt’s perturbation theory for positive C_{0}-semigroups developed in [Voi86], [Voi88]. We include the proofs for two reasons: they partly simplify the original proofs, and they demonstrate how the theory works.

Let (Ω, μ) be a measure space, 1 ⩽ p < ∞. A semigroup T of positive operators on L_{p}(μ) is called positive, which we denote by T ⩾ 0. If T_{1}, T_{2} are two positive semigroups, then T_{1} ⩽ T_{2} means T_{1}(t) ⩽ T_{2}(t) as positive operators for all t ⩾ 0. If T is a C_{0}-semigroup on L_{p}(μ) with generator −A, and V ∈ L_{∞}(μ), then T_{V} denotes the C_{0}-semigroup generated by −(A + V). For the remainder of this section let T, T_{1}, T_{2} be positive C_{0}-semigroups on L_{p}(μ).

The following two inequalities (see [Voi88; Prop. 1.3]) lie at the heart of Voigt’s perturbation theory. Most of the subsequent proofs rely on these inequalities only. Let V, V_{1}, V_{2} ∈ L_{∞}(μ) be real-valued. Then

V_{1} ⩽ V_{2} ⟹ T_{V_{1}} ⩾ T_{V_{2}} ⩾ 0,  (1.1)
V ⩾ 0, T_{1} ⩽ T_{2} ⟹ 0 ⩽ (T_{2})_{V} − (T_{1})_{V} ⩽ T_{2} − T_{1}.  (1.2)

The first statement and the first inequality of the second one follow from the Trotter product formula, T_{V} (t) = s-lim_{n→∞}(T (\frac{t}{n})e^{−tV/n})^{n} for all t ⩾ 0 (cf. [EnNa00; Exercise III.5.11]). The second inequality in (1.2) is equivalent to T_{1} − (T_{1})_{V} ⩽ T_{2} − (T_{2})_{V} which in turn follows from Duhamel’s formula, T (t) − T_{V} (t) = ∫_{0}^{t} T (t − s)V T_{V} (s) ds, since (T_{1})_{V} ⩽ (T_{2})_{V} by the first inequality.

We are going to extend the definition of T_{V} to unbounded real-valued potentials, approximating V by V^{(n)} := (V ∧ n) ∨ (−n) and letting

T_{V} (t) := s-lim_{n→∞} T_{V^{(n)}}(t) (t ⩾ 0)  (1.3)

if the limits exist. Obviously, T_{V} is a semigroup in this case, and inequalities (1.1) and (1.2) carry over to unbounded potentials whenever the corresponding limits exist. Moreover, if V ⩾ 0 or V ⩽ 0 then (T_{V^{(n)}}) is monotone by (1.1). This leads to the following definition.

Definition 1.5. ([Voi86; Def. 2.2], [Voi88; Def. 2.1], [Voi88; Def. 3.1]) Let V: Ω → [0, ∞) be measurable.

(a) If V ⩾ 0 then the limit in (1.3) exists for all t ⩾ 0 by dominated convergence. If T_{V} is strongly continuous, V is called T -admissible. In this case, T_{V^{(n)}} → T_{V}.

(b) If V ⩽ 0 then V is called T -admissible if the limit in (1.3) exists for all t ⩾ 0 and defines a C_{0}-semigroup. In this case, T_{V^{(n)}} → T_{V}.

By monotone convergence, the limit exists if and only if sup_{n∈ℕ} ||T_{V^{(n)}}(t)|| < ∞ for all t ⩾ 0. By [Voi88; Prop. 2.2], V is T -admissible if and only if sup_{0⩽t⩽1, n∈ℕ} ||T_{V^{(n)}}(t)|| < ∞.

(c) If V ⩾ 0 and V is T -admissible then −V is T_{V} -admissible since T_{V} ⩽ (T_{V})_{−V ∧n} ⩽ (T_{V ∧n})_{−V ∧n} = T for all n ∈ ℕ by (1.1) and (1.2). If T = (T_{V})_{−V}, then V is called T -regular.

In the subsequent proofs we will make use of inequalities (1.1) and (1.2) without further notice. The crucial result concerning the notion of admissibility is as follows.

Proposition 1.6. ([Voi88; Thm. 2.6]) Let U, V ⩾ 0 be measurable. Assume that −U and V are T -admissible. Then V is T_{−U}-admissible, −U is T_{V} -admissible, and (T_{V})_{−U} = (T_{−U})_{V} = s-lim_{n,m→∞} T_{V ∧n−U∧m} = T_{V −U}.

Proof. First observe that the semigroup (T_{−U})_{V} is strongly continuous since T_{V} ⩽ (T_{−U})_{V} ⩽ T_{−U}, i.e., V is T_{−U}(·)-admissible. The fact that −U is T_{V} (·)-admissible will be shown simultaneously with (T_{V})_{−U} = (T_{−U})_{V}.

Let U_{n} := U ∧ n, V_{n} := V ∧ n (n ∈ ℕ). For m ∈ ℕ we have

0 ⩽ (T_{V_{n}})_{m−U_{m}} − (T_{V})_{m−U_{m}} ⩽ T_{V_{n}} − T_{V} → 0 (n → ∞)

and hence (T_{−U_{m}})_{V_{n}} = (T_{V_{n}})_{−U_{m}} → (T_{V})_{−U_{m}} (n → ∞). Further,

0 ⩽ (T_{−U})_{V_{n}} − (T_{−U_{m}})_{V_{n}} ⩽ T_{−U} − T_{−U_{m}} (n, m ∈ ℕ),

which implies the second equality. Letting n → ∞ we obtain

0 ⩽ (T_{−U})_{V} − (T_{V})_{−U_{m}} ⩽ T_{−U} − T_{−U_{m}},

so that (T_{V})_{−U_{m}} → (T_{−U})_{V} as m → ∞. This shows that −U is T_{V} (·)-admissible and (T_{V})_{−U} = (T_{−U})_{V}. To prove the last equality it suffices to note that

T_{V_{2n}−U_{n}} ⩽ T_{(V −U)^{(n)}} ⩽ T_{V_{n}−U_{2n}} (n ∈ ℕ)

since V_{2n} − U_{n} ⩾ (V − U)^{(n)} ⩾ V_{n} − U_{2n}.

Lemma 1.7. Let V ⩾ 0 be measurable, T_{1} ⩽ T_{2}.

(a) (cf. [LiMa97; Prop. 1.4(a), Prop. 1.5]) V is T_{1}-admissible if and only if V is T_{2}-admissible.

(b) ([LiMa97; Cor. 1.15]) If V is T_{2}-regular then V is T_{1}-regular.

Proof. (a) If V is T_{1}-admissible then (T_{2})_{V} is strongly continuous since (T_{1})_{V} ⩽ (T_{2})_{V} ⩽ T_{2}. If V is T_{2}-admissible then 0 ⩽ T_{1} − (T_{1})_{V} ⩽ T_{2} − (T_{2})_{V} implies that T_{1}(t) − (T_{1})_{V} (t) → 0 strongly as t → 0, hence (T_{1})_{V} is strongly continuous.

(b) V is T_{1}-admissible by (a). With V_{n} := V − V ∧ n we have (T_{V})_{−V ∧n} = T_{V_{n}} by Proposition 1.6. Therefore, 0 ⩽ T_{1} − (T_{1})_{V_{n}} ⩽ T_{2} − (T_{2})_{V_{n}} implies that ((T_{1})_{V} )_{−V ∧n} → T_{1} as n → ∞.

The converse of (b) is not true in general, but we have the following result.

Lemma 1.8. Let U, V ⩾ 0 be measurable.

(a) ([Voi88; Prop. 3.4]) If −U is T -admissible and V is T -regular, then V is T_{−U}-regular.

(b) (cf. [LiMa97; Cor. 1.16]) Assume that U is T -regular. Then V is T - regular if and only if V is T_{U}-regular.

Proof. (a) For m ∈ ℕ, V is T_{m−U∧m}-regular by Lemma 1.7(b) and hence T_{−U∧m}- regular. Further,

0 ⩽ T_{−U} − (T_{−U})_{V −V ∧n} ⩽ (T_{−U} − T_{−U∧m}) + (T_{−U∧m} − (T_{−U∧m})_{V −V ∧n})

for all n, m ∈ ℕ. The assertion follows by choosing first m and then n large enough.

(b) follows directly from (a) and Lemma 1.7(b) since (T_{U})_{−U} = T.

Another important application of Proposition 1.6 is the next result which, roughly speaking, expresses that negative admissible potentials are always regular.

Lemma 1.9. (cf. [Voi88; Prop. 3.3(b)]) Let V ⩾ 0 be measurable. If −V is T -admissible, then (T_{−V})_{V} = T, and V is T -regular.

Proof. Since T ⩽ (T_{−V})_{V} ⩽ T_{−V}, the semigroup (T_{−V})_{V} is strongly continuous, i.e., V is T_{−V} -admissible. Lemma 1.7(a) implies that V is T -admissible, hence (T_{V})_{−V} = (T_{−V})_{V} = T by Proposition 1.6.

In the last two results of this section, let V ⩾ 0 be measurable, and T_{p}, T_{q} consistent positive C_{0}-semigroups on L_{p}(μ), L_{q}(μ), respectively, for some p, q ∈ [1, ∞).

Lemma 1.10. ([Voi86; Prop. 3.1]) (a) (T_{p})_{V} and (T_{q})_{V} are consistent, and V is T_{p}-admissible if and only if V is T_{q}-admissible.

(b) If −V is T_{p}- and T_{q}-admissible, then (T_{p})_{−V} and (T_{q})_{−V} are consistent.

(c) V is T_{p}-regular if and only if V is T_{q}-regular.

Proof. (a) First, observe that (T_{p})_{V ∧n}, (T_{q})_{V ∧n} are consistent (n ∈ ℕ) by the Trotter product formula. Therefore, (T_{p})_{V}, (T_{q})_{V} are consistent as limits of consistent semigroups. By [Voi92], (T_{p})_{V} is strongly continuous if and only if (T_{q})_{V} is strongly continuous (since (T_{p})_{V} and (T_{q})_{V} are positive semigroups).

(b) is proved in the same way as (a).

(c) follows from (a) and (b) since −V is admissible with respect to (T_{p})_{V} as well as (T_{q})_{V}.

Corollary 1.11. Assume that −V is T_{p}-admissible. Then −V is T_{q}-admissible if and only if (T_{p})_{−V} extrapolates to a C_{0}-semigroup T̂_{q} on L_{q}(μ), and T̂_{q} = (T_{q})_{−V} in this case.

Proof. If −V is T_{q}-admissible then the semigroups (T_{p})_{−V} and (T_{q})_{−V} are consistent, by Lemma 1.10(b). This shows the “only if” part, with T̂_{q} = (T_{q})_{−V}. To show the other implication, assume that (T_{p})_{−V} extrapolates to a C_{0}-semigroup T̂_{q} on L_{q}(μ). By Lemma 1.9 and Lemma 1.10(a), T_{p} = ((T_{p})_{−V} )_{V} and (T̂_{q})_{V} are consistent semigroups, i.e., (T̂_{q})_{V} = T_{q}. This shows the T_{q}-admissibility of −V, and T̂_{q} = (T_{q})_{−V} follows from the consistency of (T_{p})_{−V} and (T_{q})_{−V}.

1.4 The first Beurling-Deny criterion for sesquilinear forms

In Section 1.2 we recalled the well-known fact that with every densely defined closed sectorial form in a Hilbert space one can associate an analytic semigroup on H (Kato’s first representation theorem). In this section we are going to associate a positive C_{0}-semigroup on L_{p}(μ) with a sesquilinear form in L_{2}(μ) fulfilling the first Beurling-Deny criterion ((Ω, μ) a measure space), even in cases when the form is not bounded from the left. This is performed in Proposition 1.19 and Definition 1.20 below.

The contents of the present section are partly new. This section, together with the previous one, is fundamental for the understanding of Chapter 3.

Definition 1.12. Let τ be a sesquilinear form in L_{2}(μ).

(a) τ is called real if Re u ∈ D(τ) for all u ∈ D(τ), and τ(u, v) ∈ ℝ for all real-valued u, v ∈ D(τ).

(b) τ is said to fulfil the first Beurling-Deny criterion if τ is real and u^{+} ∈ D(τ), τ(u^{+}, u^{−}) ⩽ 0 for all real-valued u ∈ D(τ).

The following proposition, due to E.-M. Ouhabaz, shows the relevance of these two notions.

Proposition 1.13. ([Ouh92b; Prop. 2.2 and Thm. 2.4]) Let τ be a densely defined closed sectorial form in L_{2}(μ), T the associated analytic semigroup on L_{2}(μ). Then T is real (i.e., all semigroup operators are real) if and only if τ is real, and T is positive if and only if τ fulfils the first Beurling-Deny criterion.

The next lemma states that it suffices to verify the conditions of Definition 1.12 on a form core.

Lemma 1.14. Let τ be a closable sectorial form. If τ fulfils the first Beurling-Deny criterion then so does \overline{τ}.

Proof. We first show that \overline{τ} is real. Without restriction Re τ ⩾ 0. Then

τ(Re u) ⩽ τ(Re u) + τ(Im u) = Re τ(u) (u ∈ D(τ))

since τ is real. From this we easily deduce: if u ∈ D(\overline{τ}), (u_{n}) ⊆ D(τ) with u_{n} → u in D(\overline{τ}), then Re u ∈ D(\overline{τ}) and Re u_{n} → Re u in D(\overline{τ}). By the latter we show that \overline{τ}(u, v) ∈ ℝ for all real-valued u, v ∈ D(\overline{τ}), i.e., \overline{τ} is real.

From the above it follows that the set of all real-valued elements of D(τ) is dense in the set of all real-valued elements of D(\overline{τ}). Now, for real-valued u ∈ D(τ) we have \overline{τ}(u^{+}, u − u^{+}) = −\overline{τ}(u^{+}, u^{−}) ⩾ 0 and \overline{τ}(u − u^{+}, u^{+}) = −\overline{τ}((−u)^{+}, (−u)^{−}) ⩾ 0. Thus, we can apply [MaRö92; Lemma I.4.9] to conclude that u^{+} ∈ D(\overline{τ}), \overline{τ}(u^{+}, u^{−}) ⩽ 0 for all real-valued u ∈ D(\overline{τ}).

For the remainder of this section let τ be a densely defined sesquilinear form in L_{2}(μ) fulfilling the first Beurling-Deny criterion. The next result characterises admissibility of potentials via form conditions, in the case of symmetric forms.

Proposition 1.15. (cf. [Voi86; Prop. 5.7, Prop. 5.8(a)]) Let τ be symmetric and closed, T the associated positive C_{0}-semigroup on L_{2}(μ), V: Ω → [0, ∞) measurable.

(a) The potential V is T -admissible if and only if τ + V is densely defined, and τ + V ↔ T_{V} in this case.

(b) The potential −V is T -admissible if and only if V ⩽ τ +ω for some ω ∈ ℝ. In this case, τ − V is closable and \overline{τ − V} ↔ T_{−V}.

Proof. All the assertions of the proposition, except for the closability of τ −V, are shown in [Voi86]. There the proof is given for the case of the diffusion semigroup on ℝ^{N} only, but literally the same proof carries over to the general case. The closability of τ − V is due to A. Manavi ([Man01; Prop. 12.1.7]); we present his argument here.

Note that T_{−V} is a symmetric C_{0}-semigroup. Let τ̃ be the densely defined, closed symmetric form in L_{2}(μ) associated with T_{−V}. By part (a) of the proposition we have τ̃ + V ↔ (T_{−V})_{V} = T ↔ τ, taking into account Lemma 1.9 and the definition of T. Hence τ̃ + V = τ. Since Q(V) ⊇ D(τ), this implies that τ̃ ⊇ τ − V, i.e., τ − V has a closed extension.

Proposition 1.15(a) is valid even for sectorial forms, see [Man01; Kor. 12.1.4(a)]. Part (b), however, is not valid for sectorial forms τ: the inequality V ⩽ Re τ + ω still implies that −V is T -admissible ([Man01; Prop. 12.1.11]), but the converse is not true as we will see in Example 3.31 in Section 3.4.

It is clear that a sesquilinear form τ fulfils the first Beurling-Deny criterion if and only if the same holds for τ + V, for some measurable function V: Ω → ℝ with Q(V) ⊇ D(τ). Surprisingly, a similar result holds for closability. It is a direct consequence of Proposition 1.15(b).

Corollary 1.16. (cf. [Man01; Kor. 12.1.14]) Let τ be sectorial. Then τ is closable if and only if τ + V is closable for some measurable function V ⩾ 0 with Q(V) ⊇ D(τ).

Proof. Without restriction τ is symmetric. Let V ⩾ 0 be measurable with Q(V) ⊇ D(τ). If τ is closable then it is clear that τ + V is closable. If τ + V is closable then V ⩽ \overline{τ + V} + ω for some ω ∈ ℝ. Proposition 1.15(b) implies that \overline{τ + V} − V is closable. Thus, τ is closable since τ ⊆ \overline{τ + V} − V.

Definition 1.17. Let τ be sectorial and closable, V ⩾ 0 measurable. We say that V is τ-regular if D(τ + V) is a core for τ, i.e., D(τ) ∩ Q(V) is dense in D(τ).

Obviously, if V is τ-regular then V is \overline{τ}-regular, but the converse is not true in general (D(τ + V) may be \{0\} although V is \overline{τ}-regular, see [StVo85]). The following lemma states in particular that form regularity implies semigroup regularity.

Lemma 1.18. Let τ be sectorial and closable, T the positive C_{0}-semigroup associated with \overline{τ}, V ⩾ 0 τ-regular. Then V is T -regular, and T_{V} ↔ \overline{τ + V}.

Proof. Note that \overline{τ + V} fulfils the first Beurling-Deny criterion, by Lemma 1.14. Let T_{1} be the positive C_{0}-semigroup associated with \overline{τ + V}.

Since D(τ+V) is a core for \overline{τ} and (τ+V −V ∧n)(u) → \overline{τ}(u) for all u ∈ D(τ+V), we can use [Kat80; Thm. VIII.3.6] to obtain (T_{1})_{−V ∧n} → T. Thus, −V is T_{1}- admissible, and (T_{1})_{−V} = T. Lemma 1.9 implies that V is T_{1}-regular and that T_{1} = T_{V}. The latter shows the second assertion, and V is regular with respect to T = (T_{1})_{−V}, by Lemma 1.8(a).

In [Man01; Kor. 12.1.4(b)] it is shown that form regularity and semigroup regularity are actually equivalent, but we do not need this fact here.

Now we are ready to formulate the main result of this section which is fundamental for Chapter 3.

Proposition 1.19. Let U ⩾ 0 be measurable, Q(U) ⊇ D(τ), τ + U sectorial and closable, \overline{τ + U} ↔ T_{U,2}. Let V ⩾ 0 be (τ + U)-regular, τ + V sectorial and closable, \overline{τ + V} ↔ T_{V,2}. Let p ∈ [1, ∞).

Assume that T_{U,2} extrapolates to a positive C_{0}-semigroup T_{U,p} on L_{p}(μ) and that −U is T_{U,p}-admissible. Then the same holds with V in place of U, V is (T_{U,p})_{−U}-regular, and (T_{U,p})_{−U} = (T_{V,p})_{−V}.

Proof. Let T_{p} := (T_{U,p})_{−U}. It suffices to show that V is T_{U,p}-regular and that T_{V,2}, (T_{p})_{V} are consistent: then V is T_{p}-regular by Lemma 1.8(a) and thus (T_{U,p})_{−U} = ((T_{p})_{V} )_{−V}.

The potential U is (τ + V)-regular since Q(U) ⊇ D(τ + V), and V is (τ + U)- regular by the assumptions. Lemma 1.18 implies that (T_{V,2})_{U} ↔ \overline{(τ + V) + U} = \overline{(τ + U) + V} ↔ (T_{U,2})_{V} and that U is T_{V,2}-regular. Therefore,

T_{V,2} = ((T_{V,2})_{U})_{−U} = ((T_{U,2})_{V} )_{−U}.

Moreover, V is T_{U,2}-regular and hence T_{U,p}-regular by Lemma 1.10(c). Since −U is T_{U,p}-admissible we obtain by Proposition 1.6 that

(T_{p})_{V} = ((T_{U,p})_{−U})_{V} = ((T_{U,p})_{V} )_{−U}.

Now we combine the above two equalities and conclude by Lemma 1.10(a) and (b) that T_{V,2} and (T_{p})_{V} are consistent.

Proposition 1.19 leads to the following definition. Recall that τ is a densely defined sesquilinear form fulfilling the first Beurling-Deny criterion.

Definition 1.20. Let p ∈ [1, ∞). We say that τ is associated with a positive C_{0}-semigroup T_{p} on L_{p}(μ), τ ↔ T_{p} on L_{p}(μ) for short, if the following holds:

There exists U ⩾ 0 such that Q(U) ⊇ D(τ), τ + U is sectorial and closable, the positive C_{0}-semigroup T_{U,2} on L_{2}(μ) associated with \overline{τ + U} extrapolates to a C_{0}-semigroup T_{U,p} on L_{p}(μ), −U is T_{U,p}-admissible, and T_{p} = (T_{U,p})_{−U}.

According to Proposition 1.19, the semigroup T_{p} is uniquely determined by the form τ. If τ itself is sectorial and closable, we can choose U = 0. In the context of forms fulfilling the first Beurling-Deny criterion, the above definition is thus an extension of the corresponding definition of the analytic semigroup on L_{2}(μ) associated with a closed sectorial form in L_{2}(μ).

The following result is a generalisation of Lemma 1.18.

Proposition 1.21. Let p ∈ [1, ∞) and assume that τ is associated with a positive C_{0}-semigroup T_{p} on L_{p}(μ). Let U ⩾ 0 be measurable, Q(U) ⊇ D(τ), and τ + U is sectorial and closable. If V ⩾ 0 is (τ + U)-regular then V is T_{p}-regular, and τ + V ↔ (T_{p})_{V}.

Proof. First assume that V ⩾ U. Then τ + V is a closable sectorial form. Let T_{V,2} be the C_{0}-semigroup associated with \overline{τ + V}. By Proposition 1.19 we obtain that T_{V,2} extrapolates to a C_{0}-semigroup T_{V,p} on L_{p}, (T_{V,p})_{−V} = T_{p}, and V is T_{p}-regular. Lemma 1.9 implies that T_{V,p} = (T_{p})_{V}, i.e., τ + V ↔ (T_{p})_{V}.

In the general case we apply the above argument to U + V in place of V. We conclude that (τ + V) + U ↔ (T_{p})_{U+V} and that U + V is T_{p}-regular. Thus, V is T_{p}-regular, by [Voi88; Prop. 3.3(a)]. Moreover, −U is (T_{p})_{U+V} (·)-admissible and ((T_{p})_{U+V} )_{−U} = (T_{p})_{V} by [Voi88; Thm. 3.4]. Hence τ + V ↔ (T_{p})_{V}.

The next proposition deals with consistent semigroups and the adjoint semigroup. Part (a) is similar to Corollary 1.11.

Proposition 1.22. Let p ∈ [1, ∞) and assume that τ is associated with a positive C_{0}-semigroup T_{p} on L_{p}(μ).

(a) Let q ∈ [1, ∞) and T_{q} a positive C_{0}-semigroup on L_{q}(μ). Then τ is associated with T_{q} if and only if T_{p}, T_{q} are consistent. In this case, τ is associated with a family of consistent C_{0}-semigroups T_{s} on L_{s}(μ), s ∈ [p ∧ q, p ∨ q].

(b) If p > 1 and T_{p}^{∗} denotes the adjoint semigroup on L_{p′}(μ) then the form τ^{∗} is associated with T_{p}^{∗}.

Note that, since T_{p} is a real semigroup, it makes no difference whether the adjoint semigroup is taken with respect to the bilinear or with respect to the sesquilinear duality bracket.

Proof of Proposition 1.22. Let U ⩾ 0 be such that Q(U) ⊇ D(τ), τ + U is sectorial and closable, the positive C_{0}-semigroup T_{U,2} on L_{2}(μ) associated with \overline{τ + U} extrapolates to a C_{0}-semigroup T_{U,p} on L_{p}(μ), −U is T_{U,p}-admissible, and T_{p} = (T_{U,p})_{−U}.

(a) Assume that τ is associated with T_{q}. Then Proposition 1.19 implies that T_{U,2} extrapolates to a positive C_{0}-semigroup T_{U,q} on L_{q}(μ), that −U is T_{U,q}- admissible, and T_{q} = (T_{U,q})_{−U}. The semigroups T_{U,p}, T_{U,q} are consistent, so T_{p}, T_{q} are consistent by Lemma 1.10(b).

Conversely, assume that T_{p}, T_{q} are consistent. Then (T_{p})_{U}, (T_{q})_{U} are consistent by Lemma 1.10(a). From Lemma 1.9 we know that

(T_{p})_{U} = ((T_{U,p})_{−U})_{U} = T_{U,p}.

Since T_{U,2}, T_{U,p} are consistent, we conclude that T_{U,2} extrapolates to the semigroup (T_{q})_{U} on L_{q}(μ). The potential U is T_{p}-regular by Proposition 1.19 and hence T_{q}- regular by Lemma 1.10(c). Thus, −U is (T_{q})_{U}-admissible, and ((T_{q})_{U})_{−U} = T_{q}. By Definition 1.20 this shows that τ is associated with T_{q}.

The last assertion now follows by Riesz-Thorin interpolation.

(b) It is easy to see that τ^{∗} + U is closable, fulfils the first Beurling-Deny criterion, and that \overline{τ^{∗} + U} = \overline{τ + U}^{∗}. Thus, \overline{τ^{∗} + U} is associated with the positive C_{0}-semigroup T_{U,2}^{∗} which in turn extrapolates to the semigroup T_{U,p}^{∗} on L_{p′}(μ). Moreover, ((T_{U,p}^{∗})_{−U∧n})_{n∈ℕ} is an increasing sequence of semigroups, and

(T_{U,p}^{∗})_{−U∧n} = ((T_{U,p})_{−U∧n})^{∗} → T_{p}^{∗} weakly as n → ∞

since (T_{U,p})_{−U∧n} → T_{p}. We deduce that (T_{U,p}^{∗})_{−U∧n} → T_{p}^{∗} strongly as n → ∞. Hence, −U is T_{U,p}^{∗}-admissible and (T_{U,p}^{∗})_{−U} = T_{p}^{∗}, i.e., τ^{∗} is associated with T_{p}^{∗}.

The following corollary shows that, in the case of symmetric forms, Definition 1.20 does not lead to new situations in which τ can be associated with a C_{0}-semigroup on L_{p}(μ).

Corollary 1.23. Let τ be symmetric.

(a) The form τ is associated with a C_{0}-semigroup on L_{2}(μ) if and only if τ is bounded below and closable.

(b) If τ is associated with a C_{0}-semigroup on L_{p}(μ), for some p ∈ [1, 2), then τ is associated with a C_{0}-semigroup on L_{q}(μ) for all q ∈ [p, p′], q ≠ ∞.

Proof. (a) The “if” part is clear, so we prove the “only if” part. Let U ⩾ 0 be such that Q(U) ⊇ D(τ), τ + U is sectorial and closable, and −U is admissible with respect to the positive C_{0}-semigroup T_{U,2} associated with \overline{τ + U}. Then, by Proposition 1.15(b), we have U ⩽ \overline{τ + U} + c for some c ∈ ℝ. This implies τ ⩾ −c since Q(U) ⊇ D(τ). Since τ + U is closable we obtain by Corollary 1.16 that τ is closable.

(b) This is a direct consequence of Proposition 1.22.

For the last result of this section recall that I is an ideal of a lattice X if u ∈ I, v ∈ X, |v| ⩽ |u| implies v ∈ I.

Lemma 1.24. Let τ be sectorial and closable, U ⩾ 0 measurable.

(a) Let D ⊆ D(τ) be a dense ideal. Then D ∩ Q(U) is dense in D(τ + U).

(b) If V ⩾ 0 is τ-regular then V is (τ + U)-regular. In particular, if U, V are τ-regular then U + V is τ-regular.

Proof. Without restriction, τ is symmetric and τ ⩾ 0.

(a) Since τ + U fulfils the first Beurling-Deny criterion it suffices to consider 0 ⩽ u ∈ D(τ + U). Let (u_{n}) ⊆ D such that u_{n} → u in D(τ) as n → ∞. Let v_{n} := (Re u_{n})^{+}. Then v_{n} ∈ D, and v_{n} → u in L_{2}(μ). Since τ fulfils the first Beurling-Deny criterion we have lim sup_{n→∞} τ(v_{n}) ⩽ lim_{n→∞} τ(u_{n}) = τ(u).

From the lower semicontinuity of τ we deduce that v_{n} → u in D(τ) as n → ∞. Moreover, τ((u − v_{n})^{+}) ⩽ τ(u − v_{n}) → 0 and thus v_{n} ∧ u = u − (u − v_{n})^{+} → u in D(τ) as n → ∞. Finally, v_{n} ∧ u ∈ D ∩ Q(U), and v_{n} ∧ u → u in Q(U) by dominated convergence.

(b) Just apply (a) to D = D(τ + V).

1.5 Dirichlet forms

In this section we introduce the notions of sub-Markovian semigroups and (nonsymmetric) Dirichlet forms, the main source being [MaRö92]. For this thesis, the main example of a Dirichlet form is the form corresponding to a homogeneous second order elliptic differential operator with real coefficients—the principal part of the type of elliptic operators we investigate in Chapter 3. At the end of the section we present a perturbation result for symmetric sub-Markovian semigroups which is due to V. Liskevich and Yu. Semenov.

Let (Ω, μ) be a measure space. An operator B ∈ L(L_{2}(μ)) is called sub-Markovian if B is positive and L_{∞}-contractive, i.e., ||Bf||_{∞} ⩽ ||f||_{∞} for all f ∈ L_{2}(μ) ∩ L_{∞}(μ). It is easy to see that B is sub-Markovian if and only if f ⩽ 1 a.e. implies that Bf ⩽ 1 a.e. for all f ∈ L_{2}(μ): for the “if” part note that f ⩾ 0 is equivalent to −nf ⩽ 1 for all n ∈ ℕ.

A C_{0}-semigroup T on L_{2}(μ) is called sub-Markovian if all semigroup operators T (t) are sub-Markovian. The next result gives a characterisation of this property for the case that T is associated with a closed sectorial form τ ⩾ 0. We state the result in a core version which we did not find in the literature, so we include a proof. We write D(τ)_{r} for the set of real-valued elements of D(τ).

Proposition 1.25. (cf. [MaRö92; Prop. I.4.3, Thm. I.4.4]) Let τ be a densely defined closed sectorial form in L_{2}(μ), τ ⩾ 0, and T the associated analytic semigroup on L_{2}(μ). Then T is sub-Markovian if and only if τ is real and

u ∧ 1 ∈ D(τ), τ(u ∧ 1, (u − 1)^{+}) ⩾ 0 (u ∈ D)  (1.4)

for some dense subset D of D(τ)_{r}.

Proof. If T is sub-Markovian then τ is real by Proposition 1.13, and (1.4) follows from [MaRö92; Prop. I.4.3 (ii) ⇒ (i), Thm. I.4.4 (iv) ⇒ (i)], for D = D(τ).

Conversely, assume that τ is real and that (1.4) holds for a dense subset D of D(τ)_{r}. We will show that

(u − 1)^{+} ∈ D(τ), τ(u, (u − 1)^{+}) ⩾ 0 (u ∈ D(τ)_{r});

then it follows that T is sub-Markovian, by [MaRö92; Prop. I.4.3 (iii) ⇒ (ii)]. Let first u ∈ D. Note that u = u ∧ 1 + (u − 1)^{+} (this in particular implies that (u − 1)^{+} ∈ D(τ) in (1.4)), hence

τ(u, (u − 1)^{+}) = τ(u ∧ 1, (u − 1)^{+}) + τ((u − 1)^{+}) ⩾ τ((u − 1)^{+}) ⩾ 0.

From this we also obtain, by the sectoriality of τ (see [Kat80; Sec. VI.3, (1.31)]), that there exists c ⩾ 1 such that

||(u − 1)^{+}||_{τ}^{2} = τ((u − 1)^{+}) + ||(u − 1)^{+}||_{2}^{2}
⩽ (τ + 1)(u, (u − 1)^{+}) ⩽ c||u||_{τ}||(u − 1)^{+}||_{τ}.

Therefore, ||(u − 1)^{+}||_{τ} ⩽ c||u||_{τ} for all real-valued u ∈ D.

Let now u ∈ D(τ)_{r}. Let (u_{n}) ⊆ D such that u_{n} → u in D(τ). Then

(u_{n} − 1)^{+} ∈ D(τ), τ(u_{n}, (u_{n} − 1)^{+}) ⩾ 0

for all n ∈ ℕ. Thus, it remains to show (u − 1)^{+} ∈ D(τ) and (u_{n} − 1)^{+} → (u − 1)^{+} weakly in D(τ). This in turn follows from (u_{n} − 1)^{+} → (u − 1)^{+} in L_{2}(μ), sup_{n∈ℕ} ||(u_{n} − 1)^{+}||_{τ} ⩽ c sup_{n∈ℕ} ||u_{n}||_{τ} < ∞ and the lower semicontinuity of Re τ.

The above proposition leads to the following definition.

Definition 1.26. A sesquilinear form τ in L_{2}(μ) is called Dirichlet form if τ is densely defined, real, sectorial and closed, Re τ ⩾ 0 and

u ∧ 1 ∈ D(τ), τ(u ∧ 1, (u − 1)^{+}) ⩾ 0, τ((u − 1)^{+}, u ∧ 1) ⩾ 0 (u ∈ D(τ)_{r}),

or equivalently, τ and τ^{∗} fulfil condition (1.4) for some dense subset D of D(τ)_{r}.

Observe that Re τ is a Dirichlet form if τ is a Dirichlet form; but it is easy to show that the converse is not true.

By duality and interpolation we immediately obtain the fundamental result about Dirichlet forms.

Theorem 1.27. Let τ be a densely defined closed sectorial form in L_{2}(μ), T the associated analytic semigroup on L_{2}(μ). Then τ is a Dirichlet form if and only if T is positive and L_{p}-contractive for all 1 ⩽ p ⩽ ∞.

Now we introduce our main example of a Dirichlet form which will be the starting point in Chapter 3. Let N ∈ ℕ, ∅ ≠ Ω ⊆ ℝ^{N} an open set and a: Ω → ℝ^{N} ⊗ ℝ^{N} a measurable matrix-valued function. If a ∈ L_{1,loc}, i.e., a_{jk} ∈ L_{1,loc}(Ω) for all 1 ⩽ j, k ⩽ N, then we can define a sesquilinear form τ in L_{2}(Ω) by

τ(u, v) := ⟨a∇u, ∇v⟩, D(τ) := C_{c}^{∞}(Ω).

Here, ⟨f, g⟩ is defined as ∫_{Ω} f(x)·\overline{g}(x) dx whenever f ·\overline{g} ∈ L_{1}(Ω), for f, g: Ω → ℂ^{N} measurable.

We say that a is sectorial (with constant α ⩾ 0) if

|Im(aζ · \overline{ζ})| ⩽ α Re(aζ · \overline{ζ}) a.e. for all ζ ∈ ℂ^{N},

or equivalently, aζ·\overline{ζ} ∈ \overline{S_{arctan α}} a.e. for all ζ ∈ ℂ^{N} (here, S_{0} := (0, ∞)). Obviously, τ is sectorial if a is sectorial. By [Vog00] the converse is also true.

In the following we assume that a is sectorial. Define the sectorial form τ_{N} in L_{2}(Ω) by

τ_{N}(u, v) := ⟨a∇u, ∇v⟩, D(τ_{N}) := \{u ∈ W_{1,loc}^{1}(Ω) ∩ L_{2}(Ω); a∇u·∇\overline{u} ∈ L_{1}(Ω)\}.

The index N indicates that the associated sectorial operator in L_{2}(Ω) corresponds to Neumann boundary conditions (in case τ_{N} is a densely defined closed sectorial form).

In order to see that τ_{N} is defined on D(τ_{N}), i.e., a∇u · ∇\overline{v} ∈ L_{1}(Ω) for all u, v ∈ D(τ_{N}), let a_{s} := \frac{a+a^{⊤}}{2} denote the symmetric part of a. By the sectoriality of a we obtain that

D(τ_{N}) = \{u ∈ W_{1,loc}^{1}(Ω) ∩ L_{2}(Ω); a_{s}∇u · ∇\overline{u} ∈ L_{1}(Ω)\}.

Moreover, by [Kat80; Sec. VI.3, (1.31)], there exists c ⩾ 1 such that

|aξ · η| ⩽ c(a_{s}ξ · \overline{ξ})^{\frac{1}{2}} (a_{s}η · \overline{η})^{\frac{1}{2}} (ξ, η ∈ ℂ^{N}).  (1.5)

Thus,

|a∇u · ∇\overline{v}| ⩽ c(a_{s}∇u · ∇\overline{u})^{\frac{1}{2}} (a_{s}∇v · ∇\overline{v})^{\frac{1}{2}} ∈ L_{1}(Ω) (u, v ∈ D(τ_{N})),

i.e., τ_{N} is defined on D(τ_{N}).

The matrix function a is called uniformly elliptic if a ∈ L_{∞}, i.e., a_{jk} ∈ L_{∞}(Ω) for all 1 ⩽ j, k ⩽ N, and there exists ε > 0 such that a ⩾ ε a.e., i.e.,

aξ · ξ = ∑_{j,k=1}^{N} a_{jk}ξ_{j}ξ_{k} ⩾ ε|ξ|^{2} a.e. for all ξ ∈ ℝ^{N}.

It is standard (see, e.g., [Dav89; Thm. 1.3.9]) that τ_{N} is a Dirichlet form if a is symmetric and uniformly elliptic. But we are going to study much more general cases. First observe that, by the chain rule (cf. [BoMu82; Thm. 4.2]), ∇(u ∧ 1) = χ_{[u<1]}∇u and ∇(u − 1)^{+} = χ_{[u>1]}∇u for all u ∈ W_{1,loc}^{1}(Ω). This implies that u ∧ 1 ∈ D(τ_{N}) and τ(u ∧ 1, (u − 1)^{+}) = τ((u − 1)^{+}, u ∧ 1) = 0 for all real-valued u ∈ D(τ_{N}). Moreover, it is easy to see that τ_{N} is real, and Re τ_{N} ⩾ 0 since a is sectorial. Therefore, τ_{N} is a Dirichlet form as soon as τ_{N} is densely defined and closed. More generally, if τ_{N} is densely defined and closable then \overline{τ_{N}} is a Dirichlet form.

If a ∈ L_{1,loc} then C_{c}^{∞}(Ω) ⊆ D(τ_{N}); in particular, τ_{N} is densely defined in this case. The following result is an easy criterion guaranteeing that τ_{N} is closed.

Proposition 1.28. (cf. [RöWi85; Thm. 3.2]) Assume that a is sectorial, a.e. invertible, and a^{−1} ∈ L_{1,loc}. Then the form τ_{N} defined above is sectorial and closed. In particular, if τ_{N} is densely defined then it is a Dirichlet form.

In the proof of this proposition we will make use of the next lemma which shows that the assumption a invertible, a^{−1} ∈ L_{1,loc} is equivalent to a_{s} invertible, a_{s}^{−1} ∈ L_{1,loc}. For a matrix a ∈ ℝ^{N} ⊗ ℝ^{N} we write |a| to denote the norm of the operator a in Euclidean space ℝ^{N}. Observe that a_{s} ⩾ 0 if a is sectorial, so the square root a_{s}^{1/2} of a_{s} exists and we obtain

Re(aζ · \overline{ζ}) = a_{s}ζ · \overline{ζ} = |a_{s}^{1/2}ζ|^{2} (ζ ∈ ℂ^{N}).  (1.6)

Lemma 1.29. Let a ∈ ℝ^{N} ⊗ ℝ^{N} be sectorial with constant α. Then |a| ⩽ (α + 1)|a_{s}|, and a is invertible if and only if a_{s} is invertible. If a is invertible then a^{−1} is sectorial with constant α, and

a_{s} ⩽ a^{⊤}a_{s}^{−1}a ⩽ (1 + α^{2})a_{s},  (1.7)
(a^{−1})_{s} ⩽ a_{s}^{−1} ⩽ (1 + α^{2})(a^{−1})_{s}.

Proof. Let ǎ := a − a_{s} denote the antisymmetric part of a. The sector condition implies that

|ǎξ · η| ⩽ α|a_{s}^{1/2}ξ| · |a_{s}^{1/2}η| (ξ, η ∈ ℝ^{N})

(cf. (1.5)). From this we deduce

|ǎξ| ⩽ α|a_{s}^{1/2}||a_{s}^{1/2}ξ| (ξ ∈ ℝ^{N}).

In particular, |ǎ| ⩽ α|a_{s}^{1/2}||a_{s}^{1/2}| = α|a_{s}| and thus |a| ⩽ (α + 1)|a_{s}|.

In order to show the second assertion, assume that a_{s} is invertible. Then aξ · ξ = |a_{s}^{1/2}ξ|^{2} > 0 and hence aξ ≠ 0 for all 0 ≠ ξ ∈ ℝ^{N}, i.e., a is invertible. Conversely, if a_{s} is not invertible then there exists 0 ≠ ξ ∈ ℝ^{N} such that |a_{s}^{1/2}ξ|^{2} = 0. By the above, |ǎξ| ⩽ α|a_{s}^{1/2}||a_{s}^{1/2}ξ| = 0 and thus aξ = a_{s}^{1/2}(a_{s}^{1/2}ξ) + ǎξ = 0, i.e., a is not invertible.

Let now a be invertible. Then the sector condition implies that

a^{−1}ζ · \overline{ζ} = a(a^{−1}\overline{ζ}) · \overline{(a^{−1}\overline{ζ})} ∈ \overline{S_{arctan α}} (ζ ∈ ℂ^{N}),

i.e., a^{−1} is sectorial with constant α. Since ǎ is antisymmetric, we have

a^{⊤}a_{s}^{−1}a = (a_{s} + ǎ^{⊤})a_{s}^{−1}(a_{s} + ǎ) = a_{s} + ǎ^{⊤} + ǎ + ǎ^{⊤}a_{s}^{−1}ǎ = a_{s} + ǎ^{⊤}a_{s}^{−1}ǎ.

Observe that ǎ^{⊤}a_{s}^{−1}ǎ = (a_{s}^{−1/2}ǎ)^{⊤}(a_{s}^{−1/2}ǎ) ⩾ 0. Thus, (1.7) is equivalent to |a_{s}^{−1/2}ǎξ| ⩽ α|a_{s}^{1/2}ξ| for all ξ ∈ ℝ^{N}. This in turn follows from

|a_{s}^{−1/2}ǎξ · η| = |ǎξ · a_{s}^{−1/2}η| ⩽ α|a_{s}^{1/2}ξ| · |η| (ξ, η ∈ ℝ^{N}).

The last assertion is a direct consequence of (1.7), the identity a^{⊤}(a^{−1})_{s}a = a_{s}, and the following elementary observation. Let b, c ∈ ℝ^{N} ⊗ ℝ^{N} be symmetric. Then

b ⩽ c if and only if a^{⊤}ba ⩽ a^{⊤}ca

since a is invertible.

By the functional calculus one can show that for a continuous function f: ℝ → ℂ the mapping B ↦ f(B), in the space of all selfadjoint operators B ∈ L(ℝ^{N}), is continuous. We infer that the function x ↦ a_{s}(x)^{1/2} on Ω is measurable. By (1.6) we obtain that

D(τ_{N}) = {u ∈ W_{1,loc}^{1}(Ω) ∩ L_{2}(Ω); a_{s}^{1/2}∇u ∈ L_{2}(Ω)^{N}}

and that the norm in D(τ_{N}) is given by ||u||_{τ_{N}}^{2} = ||u||_{2}^{2} + ||a_{s}^{1/2}∇u||_{2}^{2}.

Proof of Proposition 1.28. First we show a_{s}^{−1/2} ∈ L_{2,loc}. Observe that a_{s}^{−1} ∈ L_{1,loc} if and only if |a_{s}^{−1}| ∈ L_{1,loc}. Since |a_{s}^{−1/2}| = |a_{s}^{−1}|^{1/2} we obtain that a_{s}^{−1/2} ∈ L_{2,loc} if and only if a_{s}^{−1} ∈ L_{1,loc}. The latter holds by Lemma 1.29.

We only have to show that D(τ_{N}) is complete. Let (u_{n}) be a Cauchy sequence in D(τ_{N}). Then u_{n} → u in L_{2}(Ω) and a_{s}^{1/2}∇u_{n} → f in L_{2}(Ω)^{N} for some u ∈ L_{2}(Ω), f ∈ L_{2}(Ω)^{N}. Since a_{s}^{−1/2} ∈ L_{2,loc} we obtain ∇u_{n} → a_{s}^{−1/2}f in L_{1,loc}(Ω)^{N}. This implies that u ∈ W_{1,loc}^{1}(Ω), a_{s}^{1/2}∇u = f ∈ L_{2}(Ω). Therefore, u ∈ D(τ_{N}) and u_{n} → u in D(τ_{N}) as n → ∞.

Let the assumptions of Proposition 1.28 hold. If a ∈ L_{1,loc} then we can define the form τ_{D} := \overline{τ_{N}↾_{C_{c}^{∞}(Ω)}}. The index D indicates that the associated sectorial operator in L_{2}(Ω) corresponds to Dirichlet boundary conditions.

Proposition 1.30. Assume that a is sectorial, a.e. invertible, and a, a^{−1} ∈ L_{1,loc}. Then W_{∞,c}^{1}(Ω) ⊆ D(τ_{D}), and τ_{D} is a Dirichlet form.

Proof. The first assertion follows from a standard convolution argument, using a ∈ L_{1,loc} and the lower semicontinuity of Re τ_{D} (see [LiVo00; proof of Lemma B4(i)]). By Proposition 1.28, τ_{N} is a Dirichlet form. Thus, for the second assertion it suffices to show u∧1 ∈ D(τ_{D}) for all u ∈ C_{c}^{∞}(Ω). This follows from the first assertion.

For the remainder of the section, (Ω, μ) will be a measure space. Recall that a function φ: ℂ → ℂ is called a normal contraction if φ(0) = 0 and |φ(x)−φ(y)| ⩽ |x − y| for all x, y ∈ ℂ, i.e., φ is Lipschitz continuous with constant 1. Then we will also say, for all measurable u: Ω → ℂ, that the function φ ∘ u is a normal contraction of u. The crucial result on normal contractions is as follows (recall that Re τ is a symmetric Dirichlet form if τ is a Dirichlet form).

Proposition 1.31. (cf. [ReSi78; Thm. XIII.51]) Let τ be a Dirichlet form in L_{2}(μ). If u ∈ D(τ) and v is a normal contraction of u then v ∈ D(τ) and Re τ(v) ⩽ Re τ(u).

We conclude this section by a result on perturbation of Dirichlet forms by real-valued potentials which is essentially due to V. Liskevich and Yu. Semenov.

Theorem 1.32. (cf. [LiSe93; Thm. 2], [LiSe96; Thm. 3.2]) Let τ be a symmetric Dirichlet form in L_{2}(μ), V: Ω → ℝ measurable. Assume that τ + V^{+} is densely defined, and V^{−} ⩽ βτ + V^{+} + c_{β} for some β < 1, c_{β} ∈ ℝ. Let p_{±} := \frac{2}{1∓\sqrt{1−β}} (the roots of the equation \frac{4}{pp′} = β).

(a) Then τ + V is a densely defined closable symmetric form, and for all p ∈ [p_{−}, p_{+}] the associated analytic semigroup T_{V,2} on L_{2}(μ) extrapolates to a positive C_{0}-semigroup T_{V,p} on L_{p}(μ), with ||T_{V,p}(t)|| ⩽ e^{c_{β}t} (t ⩾ 0). For p ∈ (p_{−}, p_{+}), the semigroup T_{V,p} is analytic.

(b) For the generator −A_{V,p} of T_{V,p} we have

⟨A_{V,p}u, |u|^{p−1} sgn u⟩ ⩾ \overline{(\frac{4}{pp′} τ + V )}(|u|^{\frac{p}{2}} sgn u) (u ∈ D(A_{V,p})).

In particular, for p ∈ (p_{−}, p_{+}) we obtain |u|^{\frac{p}{2}} sgn u ∈ D(τ) for all u ∈ D(A_{V,p}), and

⟨A_{V,p}u, |u|^{p−1} sgn u⟩ ⩾ (\frac{4}{pp′} − β)τ(|u|^{\frac{p}{2}} sgn u) − c_{β}||u||_{p}^{p}.

Remarks 1.33. (a) In [LiSe93], [LiSe96], the above theorem is proved in the more general setting of perturbation by sub-Markovian generators, not only by potentials. But the assumption on the perturbation is slightly more restrictive, namely (expressed for perturbation by a potential) V_{−} ⩽ β(τ + V_{+}) + c_{β}, with V_{−}, V_{+}: Ω → [0, ∞) such that V = V_{+} − V_{−}.

(b) Note that the inequality V^{−} ⩽ βτ + V^{+} + c_{β} obviously implies βτ + V + c_{β} ⩾ 0, but the converse is not true: For example, choose V ⩽ 0 such that D(τ + V) = {0}. Then trivially τ + V ⩾ 0, but V^{−} ⩽̸ τ. Nevertheless we have the following equivalence: V^{−} ⩽ βτ + V^{+} + c_{β} if and only if βτ + V + c_{β} ⩾ 0 and Q(V^{−}) ⊇ D(τ) ∩ Q(V^{+}).

(c) For p ∈ [1, ∞), let T_{p} be the positive C_{0}-semigroup on L_{p}(μ) associated with τ. By Proposition 1.15 we have T_{V,2} = ((T_{2})_{V^{+}})_{−V^{−}}. Thus, by Corollary 1.11, part (a) of the theorem can be expressed differently: V^{−} is (T_{p})_{V^{+}}- admissible for all p ∈ [p_{−}, p_{+}]. (But V^{−} is not T_{p}-admissible in general!) By Proposition 1.22(a) we obtain yet another reformulation of part (a): for all p ∈ [p_{−}, p_{+}], the form τ + V is associated with a quasi-contractive C_{0}-semigroup on L_{p}.

(d) The following trivial example shows that the interval [p_{−}, p_{+}] obtained in Theorem 1.32 is not always significant. Let U: Ω → [0, ∞) be measurable and unbounded, τ := U, V := −\frac{1}{2}U. Then V^{−} ⩽ βτ + c holds only if β ⩾ \frac{1}{2}. But τ +V = \frac{1}{2}U is associated with a contractive C_{0}-semigroup on L_{p} for all p ∈ [1, ∞).

Nevertheless, we will show in Section 3.4 that the interval [p_{−}, p_{+}] is sharp if τ is the form corresponding to a second order elliptic differential operator: let Ω ⊆ ℝ^{N} be open, a: Ω → ℝ^{N} ⊗ ℝ^{N} measurable and symmetric, a ⩾ 0 a.e. in the matrix sense, a, a^{−1} ∈ L_{1,loc}. Let τ_{N} be the symmetric Dirichlet form given by Proposition 1.28, and τ ⊆ τ_{N} a Dirichlet form. In Theorem 3.22 we will show that τ + V is associated with a quasi-contractive C_{0}-semigroup on L_{p} if and only if V^{−} ⩽ \frac{4}{pp′} τ +V^{+} +c for some c ∈ ℝ. For a particular example see Example 3.27.

(e) In Section 2.2 we will show that, under some additional conditions on the measure space and the semigroup, the L_{p}-scale [p_{−}, p_{+}] of existence of the semigroup can be extended, but in general without quasi-contractivity outside [p_{−}, p_{+}]. The proof of this extension result (Theorem 2.10) will reveal the relevance of Theorem 1.32(b).

The proof of Theorem 1.32 relies on [LiSe96; Thm. 2.1] and the following auxiliary result which is an L_{p}-version of Proposition 1.15. Here we use the notation v_{p}(u) := |u|^{\frac{p}{2}} sgn u and w_{p}(u) := |u|^{p−1} sgn u for u: Ω → ℂ, 1 < p < ∞. Notice that u · \overline{w_{p}(u)} = v_{p}(u) · \overline{v_{p}(u)} = |u|^{p}.

Lemma 1.34. Let 1 < p < ∞, T (t) = e^{−tA} a positive contractive C_{0}-semigroup on L_{p}(μ), and 𝔥 a closed symmetric form in L_{2}(μ). Assume that

⟨Au, w_{p}(u)⟩ ⩾ 𝔥(v_{p}(u)) (u ∈ D(A)).

Let V: Ω → ℝ be measurable, with T -admissible V^{+}, and V^{−} ⩽ 𝔥 + V^{+}. Then −V^{−} is T_{V^{+}}-admissible, e^{−tA_{V}} := (T_{V^{+}})_{−V^{−}}(t) is contractive, and

⟨A_{V} u, w_{p}(u)⟩ ⩾ \overline{𝔥 + V} (v_{p}(u)) (u ∈ D(A_{V})).  (1.8)

Proof. Notice that 𝔥 + V is closable by Proposition 1.15. It suffices to study the cases V ⩾ 0, V ⩽ 0. Let first V ⩾ 0. Then we only have to show (1.8).

Let u ∈ D(A_{V}). Without restriction assume that 0 ∈ ρ(A + V ∧ n) for all n ∈ ℕ. Then u_{n} := (A + V ∧ n)^{−1}A_{V} u → u in L_{p} and hence w_{p}(u_{n}) → w_{p}(u) in L_{p′} as n → ∞. Therefore,

(𝔥 + V ∧ n)(v_{p}(u_{n})) ⩽ ⟨(A + V ∧ n)u_{n}, w_{p}(u_{n})⟩ → ⟨A_{V} u, w_{p}(u)⟩ (n → ∞).

Without restriction v_{p}(u_{n}) → v_{p}(u) a.e. and in L_{2}. By Fatou’s lemma and the lower semicontinuity of 𝔥 we infer that V |v_{p}(u)|^{2} = lim_{n→∞} |v_{p}(u_{n})|^{2} ∈ L_{1}(μ), v_{p}(u) ∈ D(𝔥) and

(𝔥 + V)(v_{p}(u)) ⩽ ⟨A_{V} u, w_{p}(u)⟩.

Now we study the case V ⩽ 0. Recall that V^{(n)} = V ∨ (−n). The assumption V^{−} ⩽ 𝔥 implies that

⟨(A + V^{(n)})u, w_{p}(u)⟩ ⩾ (𝔥 + V^{(n)})(v_{p}(u)) ⩾ 0 (u ∈ D(A), n ∈ ℕ).

Thus, by the Lumer-Phillips theorem, T_{V^{(n)}} is a contractive semigroup for all n ∈ ℕ. By [Voi88; Prop. 2.2] we infer that V is T -admissible. To show (1.8), let u ∈ D(A_{V}) and assume without restriction that 0 ∈ ρ(A + V^{(n)}) for all n ∈ ℕ. Then we have u_{n} := (A+V^{(n)})^{−1}A_{V} u → u in L_{p} as n → ∞. Note that u_{n} ∈ D(A) implies that v_{p}(u_{n}) ∈ D(𝔥) ⊆ D(\overline{𝔥 + V}). Therefore,

\overline{𝔥 + V} (v_{p}(u_{n})) ⩽ (𝔥 + V^{(n)})(v_{p}(u_{n}))
⩽ ⟨(A + V^{(n)})u_{n}, w_{p}(u_{n})⟩ → ⟨A_{V} u, w_{p}(u)⟩ (n → ∞),

and the lower semicontinuity of \overline{𝔥 + V} implies (1.8).

Proof of Theorem 1.32. Without restriction assume that c_{β} = 0 (cf. Remark 1.33(b)). By Proposition 1.15, the symmetric form \frac{4}{pp′} τ + V is closable if \frac{4}{pp′} ⩾ β, i.e. p ∈ [p_{−}, p_{+}]. In particular, τ + V is closable.

Let p ∈ [p_{−}, p_{+}], T_{p}(t) = e^{−tA_{p}} the positive C_{0}-semigroup on L_{p}(μ) associated with τ. By [LiSe96; Thm. 2.1] we have

⟨A_{p}u, w_{p}(u)⟩ ⩾ \frac{4}{pp′} τ(v_{p}(u)) (u ∈ D(A_{p})).

(In [LiSe96], this inequality was shown for σ-finite measures μ only, for the general case one should argue as in [NaVo96].) Now Lemma 1.34 implies that −V^{−} is (T_{p})_{V^{+}}-admissible, that T_{V,p} is contractive and that (b) holds. The analyticity of T_{V,p} for p ∈ (p_{−}, p_{+}) follows from the analyticity of T_{V,2} and Stein interpolation.

Extrapolation, Analyticity, and L_{p}-spectral independence

Given a measure space (M, μ) and a C_{0}-semigroup T_{q} on L_{q}(Ω) for some measurable subset Ω ⊆ M, q ∈ [1, ∞), we are going to investigate the following two problems: under which conditions does the semigroup T_{q} extrapolate to a consistent family of C_{0}-semigroups T_{p} on L_{p}(Ω) with p-independent angle of analyticity, for p from some interval in [1, ∞) containing q? Secondly, assuming T_{q} does extrapolate to consistent C_{0}-semigroups T_{p} on L_{p}(Ω) for p from some interval, when is the spectrum of the semigroup generators p-independent? The conditions on both the space M and the semigroup T_{q} will be formulated in terms of a measurable semi-metric on M.

The chapter is organised as follows. In Section 2.1 we introduce the framework and formulate our main results. In Section 2.2 we give some account to the history of the problems of L_{p}-spectral independence and analyticity of semigroups and relate our results to the existing literature. As an application we continue the study of perturbation of sub-Markovian semigroups by potentials. For our main application, the L_{p}-theory of second order elliptic differential operators, see Chapter 3.

In Section 2.3 we develop the technique of weighted estimates which constitutes a major tool in the proofs of our results. We reformulate and prove the result on extrapolation and analyticity in a more general form in Section 2.4. In Section 2.5 we prove the theorem on L_{p}-spectral independence for semigroup generators by reducing it to a theorem on L_{p}-spectral independence for bounded operators.

2.1 Framework and main results

Throughout this chapter let 1 ⩽ p_{0} < q_{0} ⩽ ∞ be fixed, and (M, μ) a σ-finite measure space with μ(M) > 0. Let d a measurable semi-metric on M, i.e., d: M × M → [0, ∞) is measurable. Then d(x, ·) is measurable for all x ∈ M since the function y ↦ (x, y) is measurable. The open ball with respect to d with centre x and radius r will be denoted by B(x, r). We assume μ(B(x, r)) < ∞ for all x ∈ M, r > 0. In the case (p_{0}, q_{0}) ≠ (1, ∞) let v_{r}(x) := μ(B(x, r)) (x ∈ M, r > 0), whereas in the case (p_{0}, q_{0}) = (1, ∞) we only assume v_{r}: M → [0, ∞) to be measurable functions satisfying μ(B(x, r)) ⩽ v_{r}(x) for all x ∈ M, r > 0 and

v_{r} ⩽ v_{R} on M (R > r > 0), v_{r}(x) ⩽ v_{r+d(x,y)}(y) (x, y ∈ M, r > 0). (2.1)

Note that (2.1) is automatically fulfilled if v_{r}(x) = μ(B(x, r)) since B(x, r) ⊆ B(y, r + d(x, y)).

Fix a measurable subset Ω ⊆ M. We tacitly assume that functions defined on Ω are extended by 0 outside Ω when considered as functions on M. In the following we consider semigroups on L_{p}(Ω), 1 ⩽ p < ∞. The reason for introducing the space M is that the functions μ(B(·, r)) on M can behave much better than the functions μ(B(·, r) ∩ Ω) on Ω. An important example for this situation is M = ℝ^{N} and an open subset Ω ⊆ ℝ^{N}.

For the problem of extrapolation and analyticity we will need two volume growth conditions,

v_{r} ⩽ c_{0}e^{c_{1}r}v_{1} on M (r > 1),  (2.2)
v_{2r} ⩽ c_{0}v_{r} on M (0 < r ⩽ \frac{1}{2}),  (2.3)

for some c_{0} ⩾ 1, c_{1} > 0. Condition (2.2) means that the volume of balls grows at most exponentially, condition (2.3) is the doubling property for small balls. The latter is known to be equivalent to

v_{R} ⩽ c_{2}(\frac{R}{r})^{N}v_{r} (0 < r < R ⩽ 1)  (2.4)

for some N > 0. (In ‘(2.3) ⟹ (2.4)’ one obtains c_{2} = c_{0}, N = log_{2} c_{0}.)

In the case M = ℝ^{N}, d the supremum metric (this will turn out to be convenient) and μ the Lebesgue measure, conditions (2.2) and (2.3) are trivially fulfilled with v_{r}(x) = μ(B(x, r)) = (2r)^{N}. If M is a complete Riemannian manifold with Ricci curvature bounded below, d the Riemannian distance and μ the Riemannian volume, then (2.2) and (2.4) hold for v_{r}(x) = μ(B(x, r)) and N the dimension of M, by Bishop’s comparison principle (see, e.g., [GHL90; Thm. 4.19]). In the latter case, v_{r} is a function heavily depending on the space variable: in contrast to the flat space case it is not bounded below in general.

In order to formulate our main results we need the following notation. By means of the semi-metric d, we define weight functions ρ_{γ,y} on M,

ρ_{γ,y}(x) := e^{−γd(x,y)} (x, y ∈ M, γ ∈ ℝ).

Let B be a linear operator in L_{1}(M) + L_{∞}(M), and 1 ⩽ p ⩽ q ⩽ ∞. We denote the norm of B as an operator from L_{p} to L_{q} by

||B||_{p→q} := sup{||Bf||_{q}; f ∈ L_{p}(M) ∩ D(B), ||f||_{p} ⩽ 1} ∈ [0, ∞],

30 CHAPTER 2. ANALYTICITY AND L_{P} -SPECTRAL INDEPENDENCE

and for γ ∈ ℝ we define the weighted operator norm

||B||_{p→q,γ} := sup_{y∈M} ||ρ_{γ,y}Bρ_{γ,y}^{−1}↾_{{f∈L_{p}(M); ρ_{γ,y}^{−1}f∈D(B)}}||_{p→q}

= inf{c ⩾ 0; ∀ f ∈ D(B), y ∈ M: ||ρ_{γ,y}Bf||_{q} ⩽ c||ρ_{γ,y}f||_{p}} ∈ [0, ∞].

We will call an estimate of the type ||B||_{p→q,γ} < ∞ a weighted p→q-estimate or weighted norm estimate.

Theorem 2.1. Assume that (M, d) is separable and that (2.2) and (2.3) hold. Let p_{0} ⩽ s ⩽ q_{0} and T_{s} a C_{0}-semigroup on L_{s}(Ω). Assume that there exist m > 1, t_{0} > 0 and α_{0}, β_{0} ⩾ 0 with α_{0} + β_{0} = p_{0}^{−1} − q_{0}^{−1} such that

sup_{0<t⩽t_{0}} ||v_{t^{1/m}}^{α_{0}}T_{s}(t)v_{t^{1/m}}^{β_{0}}||_{p_{0}→q_{0},t^{−1/m}} < ∞.  (2.5)

Then T_{s} extrapolates to a consistent family of C_{0}-semigroups T_{p} on L_{p}(Ω), p ∈ [p_{0}, q_{0}] \ {∞}, with angle of analyticity not depending on p.

Moreover, there exist C > 0, ω ∈ ℝ, ν ⩾ 0 such that

||v_{t^{1/m}}^{α}T_{s}(t)v_{t^{1/m}}^{β}||_{p→q,γ} ⩽ Ce^{ωt+νγ^{m}t} (t > 0, γ ⩾ 0)  (2.6)

for all p_{0} ⩽ p ⩽ q ⩽ q_{0}, α, β ⩾ 0 with α + β = p^{−1} − q^{−1}.

By the phrase ‘angle of analyticity not depending on p’ we mean the following. If one of the semigroups T_{p} is analytic of angle θ, then all of them are analytic of angle θ; if one of the semigroups is not analytic, then none of them is analytic. We point out that the above theorem contributes to the solution of two problems, extrapolation as well as analyticity of semigroups. So to say, it deals with extension of the L_{p}-scale as well as extension of the time scale.

For the case of Euclidean space ℝ^{N} we immediately obtain, recalling v_{r} = (2r)^{N}:

Corollary 2.2. Let Ω ⊆ ℝ^{N} be measurable, p_{0} ⩽ s ⩽ q_{0} and T_{s} a C_{0}-semigroup on L_{s}(Ω). Assume that there exist m > 1_{1}, t_{0}, C > 0 such that

||T_{s}(t)||_{p_{0}→q_{0},t^{−1/m}} ⩽ Ct^{− \frac{N}{m} ( \frac{1}{p_{0}} − \frac{1}{q_{0}} )} (0 < t ⩽ t_{0}).

Then T_{s} extrapolates to a consistent family of C_{0}-semigroups T_{p} on L_{p}(Ω), p ∈ [p_{0}, q_{0}] \ {∞}, with angle of analyticity not depending on p.

Remarks 2.3. (a) The set of estimates in (2.6) can be considered as a generalised Gaussian upper bound (cf. [Sch96; p. 44]). Notice that (2.6) trivially implies (2.5).

In the case (p_{0}, q_{0}) = (1, ∞), estimate (2.6) with p = 1, q = ∞, α = β = \frac{1}{2} is equivalent to the following Gaussian upper bound of order m on the semigroup kernel:

|k_{t}(x, y)| ⩽ C(v_{t^{1/m}}(x)v_{t^{1/m}}(y))^{− \frac{1}{2}} exp(ωt − c(\frac{d(x,y)^{m}}{t})^{\frac{1}{m−1}} ) (t > 0, x, y ∈ Ω).

The equivalence will be shown by Davies’ trick, see Proposition 2.7 below. P. Lie and S. T. Yau [LiYa86] proved that a Gaussian upper bound of order m = 2 holds for the heat semigroup on Riemannian manifolds with Ricci curvature bounded below.

(b) The measure μ is assumed to be σ-finite since this is a necessary condition for the property that all balls B(x, r) have finite volume: for fixed x_{0} ∈ M we have M = ⋃_{n∈ℕ} B(x_{0}, n).

(c) Observe that (2.2) and (2.3) imply that v_{r} > 0 for all r > 0: recall that μ(M) > 0. Thus, for all x ∈ M there exists R > 0 such that v_{R}(x) ⩾ μ(B(x, R)) > 0. By (2.2) and (2.3) we obtain v_{r}(x) > 0 for all r > 0. In the case (p_{0}, q_{0}) ≠ (1, ∞) we thus have μ(B(x, r)) > 0 for all x ∈ M, r > 0.

(d) In Lemma 2.17 below we will show the following. (M, d) is separable as soon as 0 < μ(B(x, r)) < ∞ for all x ∈ M, r > 0. In particular, (M, d) is separable if (2.2) and (2.3) hold for v_{r}(x) = μ(B(x, r)). Conversely, if M is separable then μ(B(x, r)) > 0 for all r > 0 and almost all x ∈ M. After removing a null set we can (and do) therefore assume μ(B(x, r)) > 0 for all x ∈ M, r > 0 also in the case (p_{0}, q_{0}) = (1, ∞).

For the second result which deals with L_{p}-spectral independence, the exponential volume growth condition (2.2) is too weak. In fact, it is known that in the case of exponential volume growth the L_{p}-spectrum of the semigroup generators typically does depend on p (see, e.g., [Stu93; Prop. 2(b)]). Instead, we need the following subexponential volume growth condition

∀ ε > 0 ∃ c_{ε} > 0 ∀ r > 1 : v_{r} ⩽ c_{ε}e^{εr}v_{1},  (2.7)

as in [Stu93; p. 443]. We further assume

μ(B(x, 1)) > 0 (x ∈ M).  (2.8)

Observe that, if (2.7) holds then the latter condition is automatically fulfilled in the case (p_{0}, q_{0}) ≠ (1, ∞) (cf. Remark 2.3(c)). Moreover, if (M, d) is separable then (2.8) holds (without restriction, cf. Remark 2.3(d)). Recall that v_{r}(x) = μ(B(x, r)) in the case (p_{0}, q_{0}) ≠ (1, ∞), whereas μ(B(x, r)) ⩽ v_{r}(x) in the case (p_{0}, q_{0}) = (1, ∞), for all x ∈ M, r > 0.

Theorem 2.4. Assume that (2.7) and (2.8) hold. Let T_{p}(t) = e^{−tA_{p}} be consistent C_{0}-semigroups on L_{p}(Ω), p ∈ [p_{0}, q_{0}]\{∞}. Assume that there exist C, K, t_{0}, γ_{0} > 0 and α, β ⩾ 0 with α + β = p_{0}^{−1} − q_{0}^{−1} such that

||v_{1}^{α}T_{p_{0}}(t)v_{1}^{β}||_{p_{0}→q_{0},γ_{0}} ⩽ Ct^{−K} (0 < t ⩽ t_{0}).  (2.9)

Then the spectrum σ(A_{p}) does not depend on p ∈ [p_{0}, q_{0}]\{∞}, and the operators A_{p} have consistent resolvents.

Corollary 2.5. Let Ω ⊆ ℝ^{N} be measurable, T_{p}(t) = e^{−tA_{p}} consistent C_{0}-semi-groups on L_{p}(Ω), p ∈ [p_{0}, q_{0}]\{∞}. Assume that there exist C, K, t_{0}, γ_{0} > 0 such that

||T_{p_{0}}(t)||_{p_{0}→q_{0},γ_{0}} ⩽ Ct^{−K} (0 < t ⩽ t_{0}).

Then the spectrum σ(A_{p}) does not depend on p ∈ [p_{0}, q_{0}]\{∞}, and the operators A_{p} have consistent resolvents.

Remark 2.6. Condition (2.9) is in particular fulfilled if the doubling property (2.3) holds and there exists m > 1 such that

sup_{0<t⩽t_{0}} ||v_{t^{1/m}}^{α}T_{p_{0}}(t)v_{t^{1/m}}^{β}||_{p_{0}→q_{0},γ_{0}} < ∞.

In this case, we can choose K = \frac{N}{m}(\frac{1}{p_{0}} − \frac{1}{q_{0}} ) in (2.9), with N from (2.4). Thus, if (2.3) holds then we have the following relation between the assumptions of Theorems 2.1 and 2.4: The volume growth assumption (2.7) for large balls is more restrictive than (2.2) in Theorem 2.1, whereas assumption (2.9) on the semigroup is less restrictive than (2.5) in Theorem 2.1:

In estimate (2.9) the size of the exponent −K of t does not matter. In contrast, it is important that the number N occurring in the corresponding estimate in Corollary 2.2 is the dimension of the underlying space ℝ^{N}. Moreover, in (2.9) the weighted estimate is only needed for a fixed γ_{0} whereas in (2.5) it is crucial that γ(t) = t^{− \frac{1}{m}} tends to ∞ in the right way as t → 0.

2.2 Comments and Examples

The problem of L_{p}-spectral independence for generators of consistent C_{0}-semi-groups has a long history going back to B. Simon [Sim82] where the question was posed for Schrödinger operators. The main breakthrough was made by R. Hempel and J. Voigt [HeVo86] who answered the question in the affirmative for the case that the negative part of the potential is from the Kato class. This result was a starting point for many extensions in different directions.

The three crucial properties of Schrödinger semigroups used in the proof in [HeVo86] are the following. The underlying space is Euclidean space, the semigroup has an integral kernel satisfying a Gaussian upper bound (in particular, the semigroup acts on the whole L_{p}-scale), and the semigroup on L_{2} is symmetric. W. Arendt proved in [Are94] an abstract result saying that these three conditions are already enough to ensure L_{p}-spectral independence.

In the subsequent investigations, different results were proved assuming only two of the conditions (possibly replacing the third one by another condition). K.-Th. Sturm showed that the method of [HeVo86] can be adapted to the setting of Riemannian manifolds. In [Stu93] he proved L_{p}-spectral independence for uniformly elliptic second order operators on Riemannian manifolds with Ricci curvature bounded below, assuming a volume growth condition slightly weaker than (2.7). In the setting of metric spaces with polynomially bounded volume, an abstract approach was developed by E. B. Davies in [Dav95a].

G. Schreieck and J. Voigt were the first to investigate the problem for semigroups not acting on the whole L_{p}-scale. In [ScVo94] they established L_{p}-spectral independence for Schrödinger operators on ℝ^{N} with form small negative part of the potential. In this case we have consistent C_{0}-semigroups on L_{p}(ℝ^{N}) only for p from an interval around p = 2 (see Theorem 1.32). As a result, the semigroup has no integral kernel enjoying a pointwise Gaussian upper bound. The ideas from [ScVo94] were put in a more general context in [Sch96]. A similar method was used in [Dav95b] to show L_{p}-spectral independence for higher order elliptic operators with bounded measurable coefficients. Further progress was made by Yu. Semenov [Sem97] who studied selfadjoint second order elliptic operators with unbounded coefficients in the principal part, adapting the method from [ScVo94].

In [HiSc99], again in the context of pointwise Gaussian upper bounds, the symmetry assumption on the semigroup was replaced by certain commutator estimates. P. C. Kunstmann showed in [Kun99] that the symmetry assumption can actually be dropped. Further generalisations combining the above ones can be found in [Kun00], [LiVo00] and [KuVo00]. Most of all these extensions of the result in [HeVo86] are unified in Theorem 2.4.

Most of the known results concerning the problem of analyticity are about semigroups acting on the whole L_{p}-scale. Then the question of analyticity in L_{1} is of particular interest since for 1 < p < ∞, analyticity can be shown by Stein interpolation (but with angle depending on p). In general, L_{1}-analyticity does not hold, even if the semigroup on L_{2} is symmetric and sub-Markovian (see, e.g., [Dav89; Thm. 4.3.6], [Voi96]). Starting from [Ama83], there are several specific results on certain classes of elliptic operators on domains of ℝ^{N} stating that the semigroup on L_{1} is analytic, but not giving the optimal angle ([Kat86], [CaVe88], [ArBa93], only to mention a few).

E.-M. Ouhabaz was the first to establish analyticity of angle \frac{π}{2} in L_{1}(ℝ^{N}). In his thesis ([Ouh92a]) he observed that a Gaussian upper bound on the semigroup kernel for complex times proved in [Dav89; Thm. 3.4.8] can be used to show the following. If T_{2} is a symmetric sub-Markovian semigroup on L_{2}(ℝ^{N}) satisfying a Gaussian upper bound then the corresponding consistent C_{0}-semigroups T_{p} on L_{p}(ℝ^{N}) are analytic of angle \frac{π}{2}, for all p ∈ [1, ∞). See [Ouh95] for a more general version not assuming the semigroup to be sub-Markovian.

Ouhabaz’ result was generalised in [Dav95a] from Euclidean space to metric spaces with polynomially bounded volume. Again in the context of Euclidean space, the symmetry assumption was dropped in [Hie96], with a result stating p-independence of the angle of analyticity. For a comprehensive discussion of the case (p_{0}, q_{0}) = (1, ∞) see [Are97].

Concerning the case (p_{0}, q_{0}) ≠ (1, ∞) there are few results so far, and they are restricted to Euclidean space. E. B. Davies proved the following in [Dav95b]. If H is a selfadjoint superelliptic operator on ℝ^{N} of order 2m < N, with bounded measurable coefficients, then e^{−tH} extrapolates to an analytic semigroup on L_{p}(ℝ^{N}) of angle \frac{π}{2}, for all p ∈ [\frac{2N}{N+2m}, \frac{2N}{N−2m}]. Analyticity of angle \frac{π}{2} (but not extrapolation) was also shown in [Sch96; Sec. 3.3] in a more general setting assuming a generalised Gaussian upper bound similar to (2.6). On the other hand, extrapolation was studied in [Sem00] for generalised Schrödinger semigroups with form small negative part of the potential, but without showing analyticity of angle \frac{π}{2}.

We will show that the results on L_{p}-spectral independence and analyticity discussed above are covered by Theorems 2.1 and 2.4 (and Theorem 2.26 in Section 2.5 below), except for the following. In [Stu93] a slightly weaker assumption than the subexponential volume growth condition (2.7) was used, but the proof heavily depends on the symmetry of the semigroup in L_{2} and existence on the whole L_{p}-scale. In [Kun00], [KuVo00] there are more sophisticated results concerning L_{p}-spectral independence in the case (p_{0}, q_{0}) = (1, ∞).

For the case (p_{0}, q_{0}) = (1, ∞) it is important to observe that the set of estimates in (2.6) in Theorem 2.1 is equivalent to a Gaussian upper bound of order m.

Proposition 2.7. Assume that (M, d) is separable. Let T_{2} be a C_{0}-semigroup on L_{2}(Ω). Then the estimates in (2.6) hold with α = β = \frac{1}{2}, p = 1, q = ∞ if and only if the semigroup operators T_{2}(t) have integral kernels k_{t} satisfying

|k_{t}| ⩽ C(v_{t^{1/m}} ⊗ v_{t^{1/m}})^{− \frac{1}{2}} exp(ωt − c_{m}(\frac{d^{m}}{νt})^{\frac{1}{m−1}} ) (t > 0),  (2.10)

with c_{m} = (m − 1)m^{− \frac{m}{m−1}} .

Proof. Let D ⊆ M be countable and dense, t > 0. By the Dunford-Pettis theorem, (2.6) holds if and only if T_{2}(t) has an integral kernel k_{t} satisfying

|k_{t}(x, y)| ⩽ inf_{w∈D} Cρ_{γ,w}(x)^{−1}v_{t^{1/m}}(x)^{− \frac{1}{2}} v_{t^{1/m}}(y)^{− \frac{1}{2}} ρ_{γ,w}(y)e^{ωt+νγ^{m}t}
= C(v_{t^{1/m}}(x)v_{t^{1/m}}(y))^{− \frac{1}{2}} e^{ωt+νγ^{m}t−γd(x,y)}

for almost all x, y ∈ Ω and all rational γ ⩾ 0. We now optimise with respect to γ (Davies’ trick): setting γ = (\frac{d(x,y)}{mγt})^{\frac{1}{m−1}} yields the desired conclusion.

Gaussian upper bounds are known to hold for wide classes of uniformly elliptic operators, e.g. for

(a) second order uniformly elliptic operators in divergence form on ℝ^{N} with real coefficients [Aro67], with complex coefficients in dimensions 1 and 2 [AMT98], with uniformly continuous complex coefficients in higher dimensions [Aus96];

(b) superelliptic operators of order 2m in dimensions N < 2m [Dav95b];

(c) second order uniformly elliptic operators in divergence form on Riemannian manifolds with Ricci curvature bounded below [Sal92].

For more detailed discussions of examples for which Gaussian upper bounds are valid, we refer to [HiSc99], [Kun99].

Until recently, all results concerning the case (p_{0}, q_{0}) ≠ (1, ∞) were restricted to M = ℝ^{N}, μ = λ^{N} the Lebesgue measure. The proofs relied on the ‘box method’ where ℝ^{N} is subdivided into congruent cubes Q_{j} and one works in spaces l_{r}(L_{p}(Q_{j})). In contrast, our proofs of Theorems 2.1 and 2.4 do not use the box method but rely on Lemma 2.19 as a substitute. Indeed, working in a general measure space which carries a semi-metric it is not clear what one should use instead of the partition into cubes of equal size.

The weight functions ρ_{γ,y} were not used in the context of weighted norm estimates until [KuVo00]. In [ScVo94] the functions ρ_{ξ} defined by ρ_{ξ}(x) := e^{ξx} (x, ξ ∈ ℝ^{N}) were used to prove L_{p}-spectral independence, in [Sch96] also to prove analyticity of angle \frac{π}{2}. It was set forth in [Dav95b] that (approximations of) these weight functions ρ_{ξ} are suitable for studying all three problems of interest in the present chapter: extrapolation, analyticity and L_{p}-spectral independence. For the technique of weighted estimates, which we develop in the next section, the crucial advantage of the weights ρ_{γ,y} = e^{−γ|·−y|_{∞}} is that they are integrable for γ > 0 whereas the weights ρ_{ξ} grow exponentially in direction ξ.

In [Sem97] and later in [LiVo00], the weights had the more general form ρ_{ξ} = e^{ξψ}, with an L_{1}-regular function ψ: ℝ^{N} → ℝ^{N}, i.e., ψ is Lipschitz continuous and

sup_{k∈ℤ^{N}} ∑_{j∈ℤ^{N}} e^{−|ψ(k)−ψ(j)|} < ∞.  (2.11)

With these weight functions at hand, it was possible to study elliptic operators with singular coefficients not only in the lower order terms but also in the principal part. Note that ψ(x) = x is L_{1}-regular. We will see that the more general weights are suitable for the problem of L_{p}-spectral independence only.

Before [KuVo00], all results concerning the case (p_{0}, q_{0}) ≠ (1, ∞) involved weighted operator norms of the type sup_{|ξ|=γ} ||ρ_{ξ}Bρ_{ξ}^{−1}||_{p→q} instead of the weighted norm ||B||_{p→q,γ} defined via ρ_{γ,y}. We point out that in the case of higher order elliptic operators on ℝ^{N} (see [Dav95b]), it is hard to estimate ||B||_{p→q,γ} directly since the functions ρ_{γ,y} have only one bounded weak derivative. Nevertheless we have the following result. Proposition 2.8. Let ψ: ℝ^{N} → ℝ^{N} be Lipschitz continuous, Ω ⊆ ℝ^{N} open, B: L_{∞,c}(Ω) → L_{1,loc}(Ω) a linear operator, γ > 0, 1 ⩽ p < q ⩽ ∞. Define a semi-metric d on ℝ^{N} by d(x, y) := |ψ(x) − ψ(y)|_{∞}, and let || · ||_{p→q,γ} be the corresponding weighted operator norm. Then

||B||_{p→q,γ} ⩽ 2N sup_{|ξ|=γ} ||e^{ξψ}Be^{−ξψ}||_{p→q}.

Proof. Let E := {±γe_{j}; j = 1, …, N} where e_{j} are the standard unit vectors of ℝ^{N}. Fix y ∈ ℝ^{N}, and for ξ ∈ E let ρ_{ξ} := e^{ξ(ψ−ψ(y))}. Then

ρ_{γ,y}^{−1} = e^{γ|ψ−ψ(y)|_{∞}} = max_{ξ∈E} ρ_{ξ}^{−1}.

We will make use of the fact that B has a dual operator B′: L_{∞,c}(Ω) → L_{1,loc}(Ω) and that ||B′||_{q′→p′} = ||B||_{p→q} (see [KuVo00; Lemma 10]). For f ∈ L_{∞,c}(Ω) we obtain

||ρ_{γ,y}^{−1}B′ρ_{γ,y}f||_{p′} ⩽ ∑_{ξ∈E} ||ρ_{ξ}^{−1}B′ρ_{ξ}ρ_{ξ}^{−1}ρ_{γ,y}f||_{p′} ⩽ ∑_{ξ∈E} ||ρ_{ξ}^{−1}B′ρ_{ξ}||_{q′→p′}||ρ_{ξ}^{−1}ρ_{γ,y}f||_{q′}.

Note that ρ_{ξ}^{−1}ρ_{γ,y} ⩽ 1. By duality and the definition of ρ_{ξ} we conclude that

||ρ_{γ,y}Bρ_{γ,y}^{−1}||_{p→q} = ||ρ_{γ,y}^{−1}B′ρ_{γ,y}||_{q′→p′} ⩽ ∑_{ξ∈E} ||ρ_{ξ}^{−1}B′ρ_{ξ}||_{q′→p′} ⩽ 2N max_{ξ∈E} ||e^{ξψ}Be^{−ξψ}||_{p→q}

which completes the proof.

Let us now first consider the case ψ(x) = x. Then the semi-metric d defined above is just the supremum metric, and we have λ^{N}(B(x, r)) = (2r)^{N} for all x ∈ ℝ^{N}, r > 0. Hence, the volume growth conditions of both Corollaries 2.2 and 2.5 are satisfied. Moreover, we can use Proposition 2.8 to estimate ||T (t)||_{p→q,γ}.

A typical example for an L_{1}-regular function on ℝ^{1} is as follows. Define ψ on ℝ by ψ(2n + x) = 2n + 2x^{+} for all n ∈ ℤ, |x| ⩽ 1. Then ψ↾_{[2n−1,2n]} = 2n for all n ∈ ℤ and hence λ^{N}(B(x, r)) ⩾ 1 if (x − \frac{r}{2}, x + \frac{r}{2}) ∩ [2n − 1, 2n] ≠ ∅ for some n ∈ ℤ. From this we easily see that the doubling property (2.3) does not hold, i.e., the conditions of Theorem 2.1 are not fulfilled. But we show that the conditions of Theorem 2.4 do hold for L_{1}-regular ψ:

First observe the following. If ψ is Lipschitz continuous then there exists L > 0 such that d(x, y) ⩽ L|x − y|_{∞} for all x, y ∈ ℝ^{N}. This implies that B(x, 1) ⊇ x + [−\frac{1}{L}, \frac{1}{L}]^{N}. Hence λ^{N}(B(x, 1)) ⩾ (\frac{2}{L})^{N} for all x ∈ ℝ^{N}, i.e., (2.8) holds. Now, (i) ⟹ (ii) of the following lemma shows that (2.7) holds for L_{1}- regular ψ and, by Proposition 2.8, that

||v_{1}^{α}Bv_{1}^{β}||_{p→q,γ} ⩽ (c · 2^{N})^{α+β} · 2N sup_{|ξ|=γ} ||e^{ξψ}Be^{−ξψ}||_{p→q}.

Lemma 2.9. (cf. [KuVo00; Lemma 6]) Let ψ: ℝ^{N} → ℝ^{N} be Lipschitz continuous. Define the semi-metric d on ℝ^{N} as in Proposition 2.8. Then the following are equivalent:

(i) ψ is L_{1}-regular, (ii) there exists c > 0 such that λ^{N}(B(x, r)) ⩽ c(1 + r)^{N} for all r > 0, (iii) there exists c > 0 such that λ^{N}(B(x, r)) ⩽ ce^{r/2} for all r > 0.

Proof. As above, let L > 0 such that d(x, y) ⩽ L|x − y|_{∞} for all x, y ∈ ℝ^{N}. For j ∈ ℤ^{N} let Q_{j} := j + [−\frac{1}{2}, \frac{1}{2}]^{N}.

(i)⇒(ii) (cf. [LiVo00; Appendix A]). Let r > 0 and let n ∈ ℕ with n − 1 < r ⩽ n. Then

B(x, r) = {y ∈ ℝ^{N}; ψ(y) ∈ ψ(x) + (−r, r)^{N}}
⊆ ⋃{ψ^{−1}(ψ(x) + j + [0, 1]^{N}); j ∈ {−n, …, n − 1}^{N}}.

It follows that λ^{N}(B(x, r)) ⩽ (2n)^{N} sup_{z∈ℝ^{N}} λ^{N}(ψ^{−1}(z +[0, 1]^{N})). Since (2n)^{N} ⩽ 2^{N}(1 + r)^{N} it remains to show that the supremum is finite.

To this end, let z ∈ ℝ^{N} and Q := z + [0, 1]^{N}. Let x_{0}, y ∈ ψ^{−1}(Q) and choose k, j ∈ ℤ^{N} with x_{0} ∈ Q_{k} and y ∈ Q_{j}. Then d(x_{0}, y) ⩽ 1 and d(x_{0}, k), d(y, j) ⩽ L/2, hence d(j, k) ⩽ L + 1. Therefore ψ^{−1}(Q) ⊆ ⋃{Q_{j}; j ∈ ℤ^{N}, d(j, k) ⩽ L + 1} and

λ^{N}(ψ^{−1}(Q)) ⩽ #{j ∈ ℤ^{N}; d(j, k) ⩽ L + 1} ⩽ ∑_{j∈ℤ^{N}} e^{L+1−d(j,k)}.

By the L_{1}-regularity of ψ this shows that λ^{N}(ψ^{−1}(Q)) can be estimated from above independently of the cube Q.

The implication (ii)⇒(iii) is trivial.

(iii)⇒(i). For all k ∈ ℤ^{N} we have

∑_{j∈ℤ^{N}} e^{−|ψ(k)−ψ(j)|} ⩽ ∑_{j∈ℤ^{N}} e^{L/2} ∫_{Q_{j}} e^{−|ψ(k)−ψ(y)|} dy = e^{L/2} ∫_{ℝ^{N}} e^{−d(k,y)} dy.

This shows (i) since

∫_{ℝ^{N}} e^{−d(k,y)} dy ⩽ ∑_{n=1}^{∞} e^{−(n−1)}λ^{N}(B(k, n)) ⩽ ∑_{n=1}^{∞} ce^{−(n/2−1)} < ∞.

To conclude this section we want to show how our theorems can be applied in the context of perturbation of Dirichlet forms by potentials. Let M, Ω, μ, d be as in the introduction of Section 2.1. Based on Theorem 1.32 we are going to prove the following result.

Theorem 2.10. Assume that (2.2) and (2.4) hold for v_{r}(x) = μ(B(x, r)). Let τ be a symmetric Dirichlet form in L_{2}(Ω), and assume that the associated symmetric sub-Markovian semigroup T on L_{2}(Ω) satisfies the Gaussian upper bound (2.10) with m = 2. Let V: Ω → ℝ be measurable such that τ + V^{+} is densely defined and V^{−} ⩽ βτ + V^{+} + c_{β} for some β < 1, c_{β} ∈ ℝ. Assume N > 2 in (2.4) and let p_{+} := \frac{2}{1−\sqrt{1−β}}, p_{max} := \frac{N}{N−2}p_{+}, p_{min} := p′_{max}.

(a) Then T_{V}, the analytic semigroup on L_{2}(Ω) associated with \overline{τ + V}, extrapolates to an analytic semigroup T_{V,p} on L_{p}(Ω) of angle \frac{π}{2}, for all p ∈ (p_{min}, p_{max}).

(b) If the subexponential volume growth condition (2.7) holds instead of (2.2), then the spectrum of the generators of the semigroups T_{V,p} is independent of p ∈ (p_{min}, p_{max}).

Remark 2.11. (a) It was first observed by Yu. Semenov that the L_{p}-scale [p_{−}, p_{+}] given in Theorem 1.32 can be extended: in [Sem00] he studied the form τ corresponding to a selfadjoint second order uniformly elliptic operator on ℝ^{N}. He showed that T_{V} extrapolates to an analytic semigroup on L_{p}(ℝ^{N}), for all p ∈ (p_{min}, p_{max}), but he did not obtain the (optimal) angle \frac{π}{2}.

More generally, the above theorem can be applied to the following situation. Let M be a complete Riemannian manifold with Ricci curvature bounded below, d the Riemannian distance and μ the Riemannian volume. Then (2.2) and (2.4) hold for v_{r}(x) = μ(B(x, r)) and N the dimension of M, by Bishop’s comparison principle. Let Ω ⊆ M be open, τ the form corresponding to a selfadjoint second order uniformly elliptic operator on Ω subject to Dirichlet boundary conditions. Then the associated semigroup on L_{2}(Ω) satisfies a Gaussian upper bound of order m = 2 (see [Sal92; Thm. 6.3]).

(b) For p ∈ [1, ∞), let T_{p} be the positive C_{0}-semigroup on L_{p}(Ω) associated with τ. By Proposition 1.15, the above theorem can be reformulated as follows.

Assume that (2.7) and (2.3) hold for v_{r}(x) = μ(B(x, r)) and that T_{2} satisfies (2.10) with m = 2. Let V: Ω → ℝ be measurable such that \frac{1}{β} V^{+} is T -admissible and −\frac{1}{β} V^{−} is T_{\frac{1}{β} V^{+}}-admissible, for some β < 1. Assume N > 2 in (2.4) and define p_{min}, p_{max} as above.

Then, for all p ∈ (p_{min}, p_{max}), −V^{−} is (T_{p})_{V^{+}}-admissible, ((T_{p})_{V^{+}})_{−V^{−}}(t) = e^{−tA_{V,p}} is analytic of angle \frac{π}{2}, and the spectrum σ(A_{V,p}) is independent of p ∈ (p_{min}, p_{max}).

Assume, more restrictively, that there exists α > 1 such that αV^{+} is T_{1}- admissible and −αV^{−} is (T_{1})_{αV^{+}}-admissible. Then one can show, with a similar proof, that the assertions of Theorem 2.10 hold for all p ∈ [1, ∞). In fact, only the second part of the proof given below is needed, with slight changes and additions.

(c) An interesting point about Theorem 2.10 is the following. If V^{−} ⩽ βτ + V^{+} + c for some β < 1, c ∈ ℝ then τ + V is associated with a C_{0}-semigroup on L_{p}(Ω) for all p in [\frac{2N}{N+2}, \frac{2N}{N−2}], an interval not depending on β. If one only knows V^{−} ⩽ τ +V^{+} +c for some c ∈ ℝ, but not V^{−} ⩽ βτ +V^{+} +c for any β < 1, c ∈ ℝ, then τ + V is associated with a C_{0}-semigroup on L_{2}(Ω), by Proposition 1.15(b). In this situation it is not known whether τ +V is associated with a C_{0}-semigroup on L_{p}(Ω) for some p ≠ 2.

In the proof of Theorem 2.10 we will make use of the following immediate consequence of the Stein interpolation theorem (which is in fact a ‘pre-version’ of the Stein interpolation theorem). We fix an increasing sequence (Ω_{n}) of measurable subsets of Ω that have finite d-diameter (and hence finite μ-volume) such that Ω = ⋃_{n} Ω_{n}. By L_{∞,c} we denote the space of all f ∈ L_{∞}(Ω) for which there exists n ∈ ℕ such that f = 0 a.e. on Ω \ Ω_{n}.

Lemma 2.12. Let S := {z ∈ ℂ; 0 ⩽ Re z ⩽ 1}, and F: S → L_{1}(μ) + L_{∞}(μ). Assume that ⟨F (·), f⟩ is continuous and bounded, and analytic in the interior of S, for all f ∈ L_{∞,c}. Let p_{0}, p_{1} ∈ [1, ∞]. If there exist C_{0}, C_{1} > 0 such that

||F (j + it)||_{p_{j}} ⩽ C_{j} (j = 0, 1, t ∈ ℝ)

then ||F (θ)||_{p_{θ}} ⩽ C_{0}^{1−θ}C_{1}^{θ} for all θ ∈ (0, 1), where \frac{1}{p_{θ}} = \frac{1−θ}{p_{0}} + \frac{θ}{p_{1}} .

Proof of Theorem 2.10. By symmetry, we only need to study the case p ⩾ 2. From Theorem 1.32 we already know that T_{V} extrapolates to a positive C_{0}-semi-group T_{V,p} on L_{p}(Ω), for all p ∈ [2, p_{+}]. The idea is to apply Theorems 2.1 and 2.4 with p_{0} = p ∈ [2, p_{+}) and q_{0} = rp, for all 1 < r < \frac{N}{N−2}. For that purpose we need a weighted p→rp-estimate for v_{√\overline{t}}^{\frac{1}{p} − \frac{1}{rp}} T_{V} (t). By means of Theorem 1.32 we

will show an unweighted estimate for v_{√\overline{t}}^{\frac{1}{p} − \frac{1}{rp}} T_{αV} (t) for some α > 1 and then use Lemma 2.12 to derive the desired weighted estimate.

Since (2.10) holds with m = 2 we obtain by Theorem 2.1 that

||v_{√\overline{t}}^{\frac{1}{p} − \frac{1}{q}} T (t)||_{p→q,γ} ⩽ Ce^{ωt+νγ^{2}t} (t > 0, γ ⩾ 0, 1 ⩽ p ⩽ q ⩽ ∞)  (2.12)

for some C ⩾ 1, ω ∈ ℝ, ν ⩾ 0. Without restriction assume that c_{β} = ω = 0. Our first aim is to show the following Sobolev type inequality

||v\sqrt{t}^{\frac{1}{2r′}} u||_{2r} ⩽ C_{r}(||u||_{2}^{2} + tτ(u))^{\frac{1}{2}} (u ∈ D(τ), 0 < t ⩽ 1)  (2.13)

for all 1 < r < \frac{N}{N−2}. Observe that ||u||_{2}^{2} + tτ(u) = ||(1 + tA)^{\frac{1}{2}} u||_{2}^{2}, where A is the selfadjoint operator in L_{2}(Ω) associated with τ. We thus have to show

||v\sqrt{t}^{\frac{1}{2r′}} (1 + tA)^{− \frac{1}{2}} ||_{2→2r} ⩽ C_{r}. By the spectral theorem,

(1 + tA)^{− \frac{1}{2}} = Γ(\frac{1}{2})^{−1} ∫_{0}^{∞} s^{− \frac{1}{2}} e^{−s(1+tA)}ds.

Therefore,

||v\sqrt{t}^{\frac{1}{2r′}} (1 + tA)^{− \frac{1}{2}} ||_{2→2r} ⩽ Γ(\frac{1}{2})^{−1} ∫_{0}^{∞} s^{− \frac{1}{2}} e^{−s}||v\sqrt{t}^{\frac{1}{2r′}} T (st)||_{2→2r} ds.

Note that \frac{1}{2r′} = \frac{1}{2} − \frac{1}{2r}. For s ⩾ 1 we have v\sqrt{t} ⩽ v\sqrt{st}, so (2.12) yields

||v\sqrt{t}^{\frac{1}{2r′}} T (st)||_{2→2r} ⩽ C (s ⩾ 1).

For s ⩽ 1 we estimate v\sqrt{t} ⩽ c_{2}(\begin{matrix}√ & \overline{t} \\ √ & \overline{st} \end{matrix})^{N}v\sqrt{st} by (2.4), since \sqrt{t} ⩽ 1. By (2.12) we infer that

||v\sqrt{t}^{\frac{1}{2r′}} T (st)||_{2→2r} ⩽ (c_{2}s^{− \frac{N}{2}} )^{\frac{1}{2r′}} C (s ⩽ 1).

Moreover, \frac{N}{4r′} < \frac{1}{2} by the choice of r and hence

||v\sqrt{t}^{\frac{1}{2r′}} (1 + tA)^{− \frac{1}{2}} ||_{2→2r} ⩽ Γ(\frac{1}{2})^{−1}C (c_{2} ∫_{0}^{1} s^{− \frac{1}{2} − \frac{N}{4r′}} e^{−s}ds + ∫_{1}^{∞} s^{− \frac{1}{2}} e^{−s}ds) =: C_{r}.

This proves (2.13).

Let now 2 ⩽ p < p_{+}, 1 < r < \frac{N}{N−2}. By Theorem 1.32 we know that T_{V} extrapolates to a contractive analytic semigroup T_{V,p} on L_{p}(Ω), and for the generator −A_{V,p} of T_{V,p} we have

τ(u^{\frac{p}{2}} ) ⩽ (\frac{4}{pp′} − β)^{−1}⟨A_{V,p}u, u^{p−1}⟩ (0 ⩽ u ∈ D(A_{V,p})).

Let 0 ⩽ f ∈ L_{p}(Ω), u_{t} := T_{V,p}(t)f (∈ D(A_{V,p})) (t > 0). By (2.13) we obtain

||v\sqrt{t}^{\frac{1}{pr′}} u_{t}||_{rp}^{p} = ||v\sqrt{t}^{\frac{1}{2r′}} ut^{\frac{p}{2}} ||_{2r}^{2} ⩽ C_{r}^{2}(||u_{t}||_{p}^{p} + t(\frac{4}{pp′} − β)^{−1}⟨A_{V,p}u_{t}, u_{t}^{p−1}⟩) (0 < t ⩽ 1).

Since T_{V,p} is a contractive analytic semigroup, there exists c > 0 such that ||A_{V,p}T_{V,p}(t)|| ⩽ \frac{c}{t} (t > 0). We conclude that

||v\sqrt{t}^{\frac{1}{pr′}} T_{V,p}(t)f||_{rp}^{p} ⩽ C_{r}^{2}(||f||_{p}^{p} + (\frac{4}{pp′} − β)^{−1}c||f||_{p}^{p}) (0 < t ⩽ 1),

which amounts to an unweighted p→rp-estimate.

In order to derive the weighted estimate by Lemma 2.12, observe that \frac{4}{pp′} > \frac{4}{p_{+}p_{−}} = β since p ∈ [2, p_{+}). Choose α > 1 such that αβ < \frac{4}{pp′}. Then (αV)^{−} ⩽ αβτ + (αV)^{+}, and the above implies that there exists K ⩾ 1 such that

||v\sqrt{t}^{\frac{1}{pr′}} T_{αV,p}(t)f||_{rp} ⩽ K||f||_{p} (0 < t ⩽ 1, 0 ⩽ f ∈ L_{p}(Ω)).

Let 0 < t ⩽ 1, 0 ⩽ f ∈ L_{∞,c}. Note that

T_{α(V^{+}∧m−V^{−}∧n)}(t)f ↓ T_{α(V^{+}−V^{−}∧n)}(t)f (⩽ T_{αV} (t)f) as m → ∞.

Hence there exists m_{n} ∈ ℕ (depending on f!) such that, with V_{n} := V^{+} ∧ m_{n} − V^{−} ∧ n,

||v\sqrt{t}^{\frac{1}{pr′}} T_{αV_{n}}(t)f||_{rp} ⩽ (K + 1)||f||_{p} (n ∈ ℕ).

By (2.12) we also have, noting \frac{1}{p} − \frac{1}{rp} = \frac{1}{pr′}, that

||ρ_{γ,y}v\sqrt{t}^{\frac{1}{pr′}} T (t)ρ_{γ,y}^{−1}f||_{rp} ⩽ Ce^{νγ^{2}t}||f||_{p} (y ∈ M, γ ⩾ 0).

Lemma 2.12 yields

||ρ_{\frac{α−1}{α} γ,y}v\sqrt{t}^{\frac{1}{pr′}} T_{V_{n}}(t)ρ_{\frac{α−1}{α} γ,y}^{−1}f||_{rp} ⩽ (Ce^{νγ^{2}t})^{\frac{α−1}{α}} (K + 1)^{\frac{1}{α}} ||f||_{p}.

Finally, we use T_{V_{n}}(t)g ⩾ T_{V^{+}−V^{−}∧n}(t)g ↑ T_{V} (t)g (0 ⩽ g ∈ L_{2}(Ω), n → ∞) and the positivity of T_{V} to obtain

||v\sqrt{t}^{\frac{1}{pr′}} T_{V} (t)||_{p→rp,γ} ⩽ C^{1− \frac{1}{α}} (K + 1)^{\frac{1}{α}} e^{\frac{α}{α−1} νγ^{2}t} (0 < t ⩽ 1, γ ⩾ 0).

Now we are in a position to apply Theorem 2.1 with p_{0} := p and q_{0} := rp. We obtain that T_{V,p} extrapolates to a semigroup T_{V,q} on L_{q}(Ω) which is analytic of the same angle as T_{V,p}, for all q ∈ (p, \frac{N}{N−2}p). This holds for all p ∈ [2, p_{+}), so the proof of (a) is complete. In the same way we can apply Theorem 2.4 to prove part (b) (observe that estimate (2.9) is fulfilled by Remark 2.6).

2.3 Technique of weighted estimates

In this section we provide some technical tools needed in the proofs of Theorems 2.1 and 2.4. The main goal is to show Proposition 2.16 below. Let the notation and assumptions be as in the introduction of Section 2.1. Our first result deals with norm estimates for integral operators on M.

Lemma 2.13. Let 1 ⩽ p ⩽ q ⩽ ∞. Let k: M × M → ℂ be measurable, and for s ∈ [1, ∞] define

n_{s}(k) := max(ess sup_{x∈M} ||k(x, ·)||_{s}, ess sup_{y∈M} ||k(·, y)||_{s}) ∈ [0, ∞].

(a) Let s ∈ [1, ∞] with s^{−1} + p^{−1} = 1 + q^{−1}. If n_{s}(k) is finite then k defines a bounded integral operator I_{k}: L_{p}(M) → L_{q}(M), and ||I_{k}||_{p→q} ⩽ n_{s}(k).

(b) Let r, s ∈ [p, q] with p^{−1} + q^{−1} = r^{−1} + s^{−1}, and f ∈ L_{r}(Ω). Then

||y ↦ ||k(y, ·)f||_{L_{p}(Ω)}||_{L_{q}(M)} ⩽ n_{s}(k)||f||_{r}.

Proof. (a) is well-known and can be proved by an application of Fubini’s theorem and Riesz-Thorin interpolation.

(b) In the case p = ∞ there is nothing to show. For p < ∞, (b) is equivalent to

||y ↦ ∫_{Ω} |k|^{p}(y, ·)|f|^{p}dμ||_{L_{q/p}(M)} ⩽ n_{s}(k)^{p}||f||_{r}^{p} = n_{s/p}(|k|^{p})|||f|^{p}||_{r/p}

which in turn follows from (a) since (\frac{s}{p})^{−1} + (\frac{r}{p})^{−1} = 1 + (\frac{q}{p})^{−1}.

We are going to apply Lemma 2.13 to integral kernels of the type k(x, y) = v_{r}(x)^{−α}v_{r}(y)^{−β}e^{−γd(x,y)}. The next result gives an estimate for n_{s}(k).

Lemma 2.14. Let r > 0 and assume v_{R} ⩽ c_{0}e^{c_{r}R}v_{r} for all R > r, for some c_{0} ⩾ 1, c_{r} > 0. Then

n_{s}((x, y) ↦ v_{r}(x)^{−α}v_{r}(y)^{−β}e^{−γd(x,y)}) ⩽ c_{0}^{2}(1 − e^{−γr/3})^{−1}e^{2c_{r}r}

for all 1 ⩽ s ⩽ ∞, γ ⩾ 3c_{r} and a, β ⩽ 1 with α + β = s^{−1}.

Proof. First note that assumption (2.1) implies

v_{r}(x) ⩽ v_{r+d(x,y)}(y) ⩽ c_{0}e^{c_{r}(r+d(x,y))}v_{r}(y) (x, y ∈ M),

and hence

v_{r}^{δ}(x)/v_{r}^{δ}(y) ⩽ c_{0}e^{c_{r}r+c_{r}d(x,y)} (x, y ∈ M, |δ| ⩽ 1).  (2.14)

If s = ∞ then α = −β. By (2.14) we infer, since γ ⩾ c_{r}, that

v_{r}(x)^{−α}v_{r}(y)^{−β}e^{−γd(x,y)} ⩽ c_{0}e^{c_{r}r+(c_{r}−γ)d(x,y)} ⩽ c_{0}e^{c_{r}r} (x, y ∈ M).

This proves the lemma in the case s = ∞.

Secondly, we study the case s = 1. To this end, we first estimate the integral ∫_{M} e^{−γd(x,y)}dμ(y) for all x ∈ M and γ > c_{r}. For n ∈ ℕ let K_{n} := B(x, nr) \ B(x, (n − 1)r). Due to the assumption we have μ(K_{n}) ⩽ v_{nr}(x) ⩽ c_{0}e^{c_{r}nr}v_{r}(x), and therefore

∫_{M} e^{−γd(x,y)}dμ(y) = ∑_{n=1}^{∞} ∫_{K_{n}} e^{−γd(x,y)}dμ(y) ⩽ ∑_{n=1}^{∞} c_{0}e^{c_{r}nr}v_{r}(x)e^{−γ(n−1)r}

= c_{0}e^{γr} ∑_{n=1}^{∞} e^{(c_{r}−γ)nr}v_{r}(x) = \frac{c_{0}e^{c_{r}r}}{1 − e^{(c_{r}−γ)r}} v_{r}(x).

By (2.14) we have v_{r}(x)^{−α}v_{r}(y)^{−β} ⩽ c_{0}e^{c_{r}r+c_{r}d(x,y)}v_{r}(x)^{−1} since α + β = s^{−1} = 1. For γ ⩾ 3c_{r} we conclude that

∫_{M} v_{r}(x)^{−α}v_{r}(y)^{−β}e^{−γd(x,y)}dμ(y) ⩽ c_{0}e^{c_{r}r} · \frac{c_{0}e^{c_{r}r}}{1 − e^{(c_{r}−(γ−c_{r}))r}} = \frac{c_{0}^{2}e^{2c_{r}r}}{1 − e^{(2c_{r}−γ)r}} .

The same holds with α, β interchanged, or equivalently, with x, y interchanged. This completes the proof in the case s = 1 since 2c_{r} ⩽ \frac{2}{3}γ.

Finally, the case 1 < s < ∞ will be reduced to the cases s = 1, ∞ by Lemma 2.12. (Alternatively, it could be reduced to the case s = 1 by direct computation.) Without restriction assume α ⩾ β. Then 0 ⩽ α ⩽ 1. For 0 ⩽ Re z ⩽ 1 let k_{z}(x, y) := v_{r}(x)^{−α}v_{r}(y)^{α−z}e^{−γd(x,y)}. Then |k_{z}| = k_{Re z}. Above we obtained estimates for n_{1}(k_{1+it}) = n_{1}(k_{1}) and n_{∞}(k_{it}) = n_{∞}(k_{0}) (t ∈ ℝ). Making use of Lemma 2.12 we deduce the desired estimate for n_{s}(k_{1/s}).

Let us remark that a volume growth assumption of the type v_{R} ⩽ ce^{cR}v_{r} is necessary in the above lemma: assume that the assertion of the lemma holds for s = 1, α = 1, β = 0, γ = 3c_{r}. Then we have, for almost all x ∈ M:

μ(B(x, R)) ⩽ ∫_{M} e^{γR−γd(x,y)}dμ(y) ⩽ n_{1}((x, y) ↦ v_{r}(x)^{−1}e^{−γd(x,y)})e^{γR}v_{r}(x)
⩽ c_{0}^{2}(1 − e^{−c_{r}r})^{−1}e^{2c_{r}r}e^{3c_{r}R}v_{r}(x).

For γ ⩾ 0 and C ⩾ 1 we introduce a class of weight functions on M,

P (γ, C) := {ρ: M → (0, ∞) measurable; ρ(x)/ρ(y) ⩽ Ce^{γd(x,y)} for all x, y ∈ M}.

For ρ ∈ P (γ, C) and y ∈ M we have, by the definition of ρ_{γ,y},

ρ ⩽ Cρ_{−γ,y}ρ(y) and ρ(y) ⩽ Cρ_{−γ,y}ρ.  (2.15)

Notice that, by the triangle inequality, ρ_{γ,y} ∈ P (|γ|, 1) for all γ ∈ ℝ, y ∈ M. Moreover, if v_{R} ⩽ c_{0}e^{c_{r}R}v_{r} for some r > 0 and all R > r then v_{r}^{δ} ∈ P (c_{0}e^{c_{r}r}, c_{r}) for all |δ| ⩽ 1, by (2.14).

We fix an increasing sequence (Ω_{n}) of measurable subsets of Ω that have finite d-diameter (and hence finite μ-volume) such that Ω = ⋃_{n} Ω_{n}. By L_{1,loc} we denote the set of (equivalence classes of) all measurable functions f on Ω with ||χ_{Ω_{n}}f||_{1} < ∞ for all n ∈ ℕ, and by L_{∞,c} the space of all f ∈ L_{∞}(Ω) for which there exists n ∈ ℕ such that f = 0 a.e. on Ω \ Ω_{n}. Note that the elements of P (γ, C) are multiplication operators on L_{1,loc} and L_{∞,c}.

Remark 2.15. Let p, q ∈ [1, ∞], B a bounded operator on L_{p}(Ω) and ρ_{1}, ρ_{2} ∈ L_{∞,loc}, i.e., ||χ_{Ω_{n}}ρ_{j}||_{∞} < ∞ for all n ∈ ℕ, j = 1, 2. Let D(ρ_{2}, L_{p}) denote the domain of the multiplication operator ρ_{2} on L_{p}(Ω). Then

||ρ_{1}Bρ_{2}↾_{D(ρ_{2},L_{p})}||_{p→q} = ||ρ_{1}Bρ_{2}↾_{L_{∞,c}}||_{p→q}.

This follows from an application of Fatou’s lemma.

The following result is one of the crucial tools in the proofs of Theorems 2.1 and 2.4. The archetype of this result is due to G. Schreieck and J. Voigt ([ScVo94; Prop. 3.2]; see [Sem97; Lemma 5.2] and [KuVo00; Prop. 13] for improved versions). In the present form the result is new.

Proposition 2.16. Let r > 0, and assume v_{R} ⩽ c_{0}e^{c_{r}R}v_{r} for all R > r, for some c_{0} ⩾ 1, c_{r} > 0. Further assume that μ(B(x, r)) > 0 for all x ∈ M. Let γ_{0} ⩾ 8c_{r} and α_{0}, β_{0} ⩾ 0 with α_{0} + β_{0} = p_{0}^{−1} − q_{0}^{−1}. Then, for any linear operator B: L_{∞,c} → L_{1,loc} satisfying ||v_{r}^{α_{0}}Bv_{r}^{β_{0}}||_{p_{0}→q_{0},γ_{0}} ⩽ 1 we have

||ρv_{r}^{α}Bv_{r}^{β}ρ^{−1}||_{p→q} ⩽ C^{2}c_{0}^{6}(1 − e^{−γ_{0}r/8})^{−2}e^{5γ_{0}r}

for all p_{0} ⩽ p ⩽ q ⩽ q_{0}, α, β ⩾ 0 with α+β = p^{−1} −q^{−1}, C ⩾ 1, ρ ∈ P (γ_{0}/2, C). In particular,

||v_{r}^{α}Bv_{r}^{β}||_{p→q,γ} ⩽ c_{0}^{6}(1 − e^{−γ_{0}r/8})^{−2}e^{5γ_{0}r} (|γ| ⩽ γ_{0}/2).

Observe that the assumption μ(B(x, r)) > 0 is automatically fulfilled in the case (p_{0}, q_{0}) ≠ (1, ∞) (cf. Remark 2.3(c)). For the proof of Proposition 2.16 we need some preparatory lemmas. The first one will be needed in the case (p_{0}, q_{0}) = (1, ∞). At the same time it proves Remark 2.3(d).

Lemma 2.17. Assume that there exists r > 0 such that μ(B(x, r)) > 0 for all x ∈ M. Then there exists a sequence (x_{n}) ⊆ M such that M = ⋃_{n∈ℕ} B(x_{n}, 2r). In particular, if μ(B(x, r)) > 0 for all x ∈ M, r > 0 then M is separable. Conversely, if M is separable then μ(B(x, r)) > 0 for all r > 0 and almost all x ∈ M.

Proof. Fix x_{0} ∈ M and let k ∈ ℕ. By Zorn’s lemma we can choose M_{k} ⊆ B(x_{0}, k) such that (B(x, r))_{x∈M_{k}} is a maximal family of pairwise disjoint balls with radius r and centre in B(x_{0}, k). Then M_{k} is countable since μ(B(x_{0}, k + r)) < ∞ and μ(B(x, r)) > 0 for all x ∈ M_{k}. By the maximality we have B(x_{0}, k) ⊆ ⋃_{x∈M_{k}} B(x, 2r). This proves the first assertion since M = ⋃_{k∈ℕ} B(x_{0}, k).

Assume now that M is separable. Let r > 0 and choose (x_{n}) ⊆ M such that M = ⋃_{n∈ℕ} B(x_{n}, \frac{r}{2}). Then for all n ∈ ℕ with μ(B(x_{n}, \frac{r}{2})) > 0 we obtain μ(B(x, r)) > 0 for all x ∈ B(x_{n}, \frac{r}{2}). This implies the second assertion.

Corollary 2.18. Let r > 0 and assume that μ(B(x, r)) > 0 for all x ∈ M. Let γ > 0, B: L_{∞,c} → L_{1,loc} a linear operator. If ||B||_{1→∞,γ} ⩽ 1 then B has an integral kernel k satisfying |k(x, y)| ⩽ e^{4γr−γd(x,y)} for almost all x, y ∈ Ω.

Proof. Let w ∈ M. By the Dunford-Pettis theorem, the assumption implies that the operator B_{w} := ρ_{γ,w}Bρ_{γ,w}^{−1} has an integral kernel k_{w} satisfying |k_{w}| ⩽ 1 a.e. Therefore, B has an integral kernel k satisfying

k(x, y) = ρ_{γ,w}(x)^{−1}k_{w}(x, y)ρ_{γ,w}(y) (almost all x, y ∈ Ω).

According to Lemma 2.17 choose a sequence (w_{n}) ⊆ M such that M = ⋃_{n} B(w_{n}, 2r). For all n ∈ ℕ and almost all x ∈ B(w_{n}, 2r), y ∈ Ω we obtain

|k(x, y)| = e^{γd(x,w_{n})−γd(y,w_{n})} ⩽ e^{4γr−γd(x,y)},

which concludes the proof.

The next lemma will be used in the case (p_{0}, q_{0}) ≠ (1, ∞). Here is the place where the assumption v_{r}(x) = μ(B(x, r)) enters. Though being elementary, this lemma constitutes the main trick in the proof of Proposition 2.16.

Lemma 2.19. Let r > 0 and assume v_{r}(x) = μ(B(x, r)) (x ∈ M). For γ > 0, 1 ⩽ q ⩽ ∞ and f ∈ L_{q}(Ω) we then have

||f||_{q} ⩽ e^{γr}||y ↦ ||ρ_{γ,y}v_{r}^{−1/q}f||_{L_{q}(Ω)}||_{L_{q}(M)}.

Proof. By Fubini’s theorem we have, since ρ_{γ,y}(x) = ρ_{γ,x}(y) for all x, y ∈ M,

||y ↦ ||ρ_{γ,y}v_{r}^{−1/q}f||_{L_{q}(Ω)}||_{L_{q}(M)} = ||x ↦ ||ρ_{γ,x}||_{L_{q}(M)}v_{r}(x)^{−1/q}f(x)||_{L_{q}(Ω)}.

This implies the assertion since e^{γr}||ρ_{γ,x}||_{L_{q}(M)} ⩾ ||χ_{B(x,r)}||_{L_{q}(M)} = μ(B(x, r))^{1/q} for all x ∈ Ω.

In order to present the idea of the proof of Proposition 2.16 in the case (p_{0}, q_{0}) ≠ (1, ∞), let us first show a simple variant of Proposition 2.16: let r > 0 and assume that v_{R} ⩽ c_{0}e^{c_{r}R}v_{r} for all R > r, with v_{r}(x) = μ(B(x, r)). Let 1 ⩽ p ⩽ q ⩽ ∞, γ ⩾ 3c_{r} and B: L_{∞,c} → L_{1,loc} a linear operator satisfying ||v_{r}^{−1/q}Bv_{r}^{1/p}||_{p→q,γ} ⩽ 1. Then ||ρ_{γ,y}v_{r}^{−1/q}Bf||_{q} ⩽ ||ρ_{γ,y}v_{r}^{−1/p}f||_{p} for all f ∈ L_{∞,c}, y ∈ M. By Lemma 2.19 and Lemma 2.13(b) we obtain

||Bf||_{q} ⩽ e^{γr}||y ↦ ||ρ_{γ,y}v_{r}^{−1/q}Bf||_{q}||_{L_{q}(M)}

⩽ e^{γr}||y ↦ ||ρ_{γ,y}v_{r}^{−1/p}f||_{p}||_{L_{q}(M)} ⩽ e^{γr}n_{p}((x, y) ↦ ρ_{γ,y}(x)v_{r}(x)^{−1/p})||f||_{q}.

Thus, Lemma 2.14 yields ||B||_{q→q} ⩽ c_{0}^{2}(1 − e^{−γr/3})^{−1}e^{2c_{r}r+γr}.

Proof of Proposition 2.16. First case: (p_{0}, q_{0}) = (1, ∞). By Corollary 2.18, the assumption implies that the operator v_{r}^{α_{0}}Bv_{r}^{β_{0}} has an integral kernel k satisfying

|k(x, y)| ⩽ e^{4γ_{0}r−γ_{0}d(x,y)} (almost all x, y ∈ Ω).

Therefore, the kernel h of ρv_{r}^{α}Bv_{r}^{β}ρ^{−1} satisfies

|h(x, y)| ⩽ v_{r}(x)^{α−α_{0}}v_{r}(y)^{β−β_{0}}ρ(x)ρ(y)^{−1}e^{4γ_{0}r−γ_{0}d(x,y)} (almost all x, y ∈ Ω).

Note that α_{0} − α, β_{0} − β ⩽ 1, (α_{0} − α) + (β_{0} − β) = 1 − p^{−1} + q^{−1} =: s^{−1}, and ρ(x)ρ(y)^{−1} ⩽ Ce^{\frac{γ_{0}}{2} d(x,y)}. Thus, Lemma 2.14 implies that

n_{s}(h) ⩽ Ce^{4γ_{0}r} · c_{0}^{2}(1 − e^{−γ_{0}r/6})^{−1}e^{2c_{r}r}.

By Lemma 2.13(a) we obtain the assertion since 2c_{r} ⩽ γ_{0}.

Second case: (p_{0}, q_{0}) ≠ (1, ∞). Then v_{r}(x) = μ(B(x, r)) (x ∈ M). Let ρ ∈ P (γ_{0}/2, C), f ∈ L_{∞,c}. We have to show that ||ρv_{r}^{α}Bf||_{q} ⩽ C^{2}c_{0}^{6}(1−e^{−γ_{0}r/8})^{−2}· e^{5γ_{0}r}||ρv_{r}^{−β}f||_{p}.

By Lemma 2.19 we have

||ρv_{r}^{α}Bf||_{q} ⩽ e^{2γ_{0}r}||y ↦ ||ρ_{2γ_{0},y}v_{r}^{−1/q} · ρv_{r}^{α}Bf||_{q}||_{L_{q}(M)}.  (2.16)

Let s^{−1} := q^{−1} − q_{0}^{−1} and ρ_{1} := ρv_{r}^{α−q_{0}^{−1}−α_{0}}. Then v_{r}^{−1/q} · ρv_{r}^{α} = v_{r}^{−1/s}ρ_{1}v_{r}^{α_{0}}, and by Hölder’s inequality we estimate

||ρ_{2γ_{0},y}v_{r}^{−1/q}ρv_{r}^{α}Bf||_{q} ⩽ ||ρ_{3γ_{0}/8,y}v_{r}^{−1/s}||_{s}||ρ_{5γ_{0}/8,y}ρ_{1} · ρ_{γ_{0},y}v_{r}^{α_{0}}Bf||_{q_{0}}  (2.17)

for all y ∈ M. Since 3γ_{0}/8 ⩾ 3c_{r} we can apply Lemma 2.14 to the first factor on the right hand side of (2.17) and obtain

||ρ_{3γ_{0}/8,y}v_{r}^{−1/s}||_{s} ⩽ c_{0}^{2}(1 − e^{−γ_{0}r/8})^{−1}e^{2c_{r}r} (almost all y ∈ M).

To estimate the second factor, note that (2.14) implies that v_{r}^{δ} ∈ P (c_{r}, c_{0}e^{c_{r}r}) for all |δ| ⩽ 1. Since c_{r} ⩽ γ_{0}/8, this yields ρ_{1} ∈ P (\frac{5}{8}γ_{0}, Cc_{0}e^{c_{r}r}). Using (2.15) and the assumption ||ρ_{γ_{0},y}v_{r}^{α_{0}}Bf||_{q_{0}} ⩽ ||ρ_{γ_{0},y}v_{r}^{−β_{0}}f||_{p_{0}}, we thus obtain

||ρ_{5γ_{0}/8,y}ρ_{1} · ρ_{γ_{0},y}v_{r}^{α_{0}}Bf||_{q_{0}} ⩽ ||Cc_{0}e^{c_{r}r}ρ_{1}(y) · ρ_{γ_{0},y}v_{r}^{α_{0}}Bf||_{q_{0}}
⩽ Cc_{0}e^{c_{r}r}||ρ_{1}(y) · ρ_{γ_{0},y}v_{r}^{−β_{0}}f||_{p_{0}}
⩽ C^{2}c_{0}^{2}e^{2c_{r}r}||ρ_{−5γ_{0}/8,y}ρ_{1} · ρ_{γ_{0},y}v_{r}^{−β_{0}}f||_{p_{0}}
⩽ C^{2}c_{0}^{2}e^{2c_{r}r}||ρ_{3γ_{0}/8,y}ρv_{r}^{a−q_{0}^{−1}−a_{0}−β_{0}}f||_{p_{0}}.

Due to the assumption, α − q_{0}^{−1} − α_{0} − β_{0} = p^{−1} − q^{−1} − p_{0}^{−1} − β =: −β − s_{1}^{−1}. By (2.17) we therefore conclude that

||ρ_{2γ_{0},y}v_{r}^{−1/q}ρv_{r}^{α}Bf||_{q} ⩽ C^{2}c_{0}^{4}(1 − e^{−γ_{0}r/8})^{−1}e^{4c_{r}r}||ρ_{3γ_{0}/8,y}v_{r}^{−1/s_{1}} · ρv_{r}^{−β}f||_{p_{0}}.

Inserting this into (2.16) yields, by Lemma 2.13(b) and Lemma 2.14,

||ρv_{r}^{α}Bf||_{q} ⩽ e^{2γ_{0}r}C^{2}c_{0}^{4}(1 − e^{−γ_{0}r/8})^{−1}e^{4c_{r}r}n_{s_{1}}(ρ_{3γ_{0}/8,y}(x)v_{r}(x)^{−1/s_{1}}) · ||ρv_{r}^{−β}f||_{p}
⩽ C^{2}c_{0}^{6}(1 − e^{−γ_{0}r/8})^{−2}e^{6c_{r}r+2γ_{0}r}||ρv_{r}^{−β}f||_{p}.

To complete the proof, note that 6c_{r} ⩽ γ_{0}.

We conclude this section by a lemma which will be used to show strong continuity of consistent semigroups.

Lemma 2.20. Let 1 ⩽ p < ∞ and (B_{α}) a bounded net in L(L_{p}(Ω)). Assume that C := sup_{α} ||B_{α}||_{p→p,γ} < ∞ for some γ < 0, and that there exists q > p such that B_{α}f → f in L_{q}(Ω) for all f ∈ L_{∞,c}. Then B_{α} → I strongly in L(L_{p}(Ω)).

Proof. Since (B_{α}) is bounded, it suffices to show B_{α}f → f in L_{p}(Ω) for all f ∈ L_{∞,c}. Given f, let x_{0} ∈ Ω, r_{0} > 0 such that supp f ⊆ B(x_{0}, r_{0}). For r > r_{0} we obtain (note that γ < 0!)

||B_{α}f − f||_{p} ⩽ ||(B_{α}f)↾_{B(x_{0},r)} − f||_{p} + ||(B_{α}f)↾_{B(x_{0},r)^{c}}||_{p}
⩽ μ(B(x_{0}, r))^{p^{−1}−q^{−1}}||B_{α}f − f||_{q} + ||e^{γr}ρ_{γ,x_{0}}B_{α}f||_{p}.

By the assumption we have ||ρ_{γ,x_{0}}B_{α}f||_{p} ⩽ C||ρ_{γ,x_{0}}f||_{p}. Since γ < 0 this implies that ||e^{γr}ρ_{γ,x_{0}}B_{α}f||_{p} ⩽ e^{γr}C||ρ_{γ,x_{0}}f||_{p} → 0 as r → ∞, and we conclude that B_{α}f → f in L_{p}(Ω).

2.4 Extrapolation and analyticity

In order to apply Proposition 2.16 in the proof of Theorem 2.1, we need a reformulation of our volume growth conditions: assume that (2.2) and (2.3) hold. We claim that then for all ε > 0 there exists c_{0}(ε) ⩾ 1 such that

v_{R} ⩽ c_{0}(ε)e^{(c_{1}∨ \frac{ε}{r})R}v_{r} (R > r > 0).  (2.18)

This is shown by a distinguishing three cases. If R > r ⩾ 1 then v_{R} ⩽ c_{0}e^{c_{1}R}v_{1} by (2.2), and we are done since v_{1} ⩽ v_{r}. If r < R ⩽ 1 then v_{R} ⩽ c_{2}(\frac{R}{r})^{N}v_{r} by (2.4), and there exists c_{0}(ε) > 1 such that c_{2}(\frac{R}{r})^{N} ⩽ c_{0}(ε)e^{\frac{ε}{r} R} for all R > r > 0.

Let now 0 < r ⩽ 1 ⩽ R. Then v_{R}/v_{r} = v_{R}/v_{1} · v_{1}/v_{r} ⩽ c_{0}e^{c_{1}R} · c_{2}(\frac{1}{r})^{N}. If r ⩾ \frac{ε}{2c_{1}} then we conclude that v_{R}/v_{r} ⩽ c_{0}c_{2}(\frac{2c_{1}}{ε} )^{N}e^{c_{1}R}; if r < \frac{ε}{2c_{1}} then c_{1} < \frac{ε}{2r} and hence v_{R}/v_{r} ⩽ c_{0}c_{2}e^{\frac{ε}{2r} R}(\frac{R}{r})^{N} ⩽ c_{0}(ε)e^{\frac{ε}{r} R} for some c_{0}(ε) ⩾ 1 not depending on r, R. This shows (2.18).

Conversely, assume that (2.18) holds for some ε > 0, c_{0}(ε) ⩾ 1. Then we easily show estimates of the type (2.2) and (2.3).

Remark 2.21. As a direct consequence of Proposition 2.16 we obtain the following. Assume that (2.18) holds and that (Ω, d) is separable. Let B: L_{∞,c} → L_{1,loc} be a linear operator, ε > 0. Then, with K_{ε} := (c_{0}(\frac{ε}{8}))^{6}(1 − e^{−ε/8})^{−2},

||v_{r}^{α}Bv_{r}^{β}||_{p→q,γ} ⩽ K_{ε}e^{5γ_{0}r}||v_{r}^{α_{0}}Bv_{r}^{β_{0}}||_{p_{0}→q_{0},γ_{0}}

for all p_{0} ⩽ p ⩽ q ⩽ q_{0}, r > 0, γ_{0} ⩾ \frac{ε}{r} ∨ (8c_{1}), |γ| ⩽ γ_{0}/2, and α_{0}, β_{0}, α, β ⩾ 0 with α_{0} + β_{0} = p_{0}^{−1} − q_{0}^{−1}, α + β = p^{−1} − q^{−1}.

For the proof recall from Remark 2.3(d) that the separability of (Ω, d) is essentially equivalent to μ(B(x, r)) > 0 for all x ∈ M, r > 0. Then apply Proposition 2.16 with c_{0} := c_{0}(\frac{ε}{8}) and c_{r} := c_{1} ∨ \frac{ε}{8r}, and observe that (1 − e^{−γ_{0}r/8})^{−2} ⩽ (1 − e^{−ε/8})^{−2} for all γ_{0} ⩾ 8c_{r} = \frac{ε}{r} ∨ (8c_{1}).

In the proof of Theorem 2.1 we will use the following result to pass from small times to large times.

Lemma 2.22. Let T be a semigroup on L_{p}(Ω). Assume that there exist ε, t_{0} > 0, C ⩾ 1 and m > 1 such that ||T (t)||_{p→p,γ} ⩽ C for all t ⩽ t_{0}, 0 ⩽ γ ⩽ εt^{−1/m}. Then there exist ω, ν ⩾ 0 such that

||T (t)||_{p→p,γ} ⩽ Ce^{ωt+νγ^{m}t} (t, γ ⩾ 0).

Proof. For γ = 0 the assertion is well-known. In case γ > 0 let t_{γ} := (γ/ε)^{−m} ∧t_{0}. For t ⩾ 0 choose n ∈ ℕ with (n − 1)t_{γ} ⩽ t < nt_{γ}. Then n − 1 ⩽ t/t_{γ} ⩽ ((γ/ε)^{m} + t_{0}^{−1})t and \frac{t}{n} ⩽ (γ/ε)^{−m} ∧ t_{0}, in particular γ ⩽ ε(\frac{t}{n})^{−1/m}. By the assumption it follows that

||T (t)||_{p→p,γ} ⩽ ||T (\frac{t}{n})||_{p→p,γ}^{n} ⩽ C · C^{n−1} ⩽ Ce^{ln C·((γ/ε)^{m}+t_{0}^{−1})t} = Ce^{ωt+νγ^{m}t}

with ω = t_{0}^{−1} ln C and ν = ε^{−m} ln C.

We further need to extend the weighted estimate in Lemma 2.22 from real to complex times. The next proposition serves this purpose. Comparable results are shown in [Dav89], [Sch96] and [Hie96] by means of the Phragmen-Lindelöf theorem. But it seems to be more natural to use Stein interpolation, similar to the proof of [Dav95b; Lemma 9] by means of the three lines theorem.

Proposition 2.23. Let ρ: Ω → (0, ∞) with ρ, ρ^{−1} ∈ L_{∞,loc}, and θ ∈ (0, \frac{π}{2}]. Let F: S_{θ} → L(L_{p}) be a bounded continuous function, analytic in the interior of S_{θ}, satisfying the inequality

||ρ^{γ}F (t)ρ^{−γ}|| ⩽ Ce^{νγ^{m}t} (t, γ ⩾ 0)

for some C ⩾ 1, ν > 0, m > 1. Then for all φ ∈ (0, θ) there exists ν_{φ} > 0 such that

||ρ^{γ}F (z)ρ^{−γ}|| ⩽ C_{1}e^{ν_{φ}γ^{m} Re z} (z ∈ S_{φ}, γ ⩾ 0),

with C_{1} = max\{||F ||_{∞}, C\}. If θ = \frac{π}{2} then one can choose ν_{φ} = (1 − \frac{2}{π}φ)^{−m}ν.

Proof. Fix γ ⩾ 0 and let φ(z) := exp(−\frac{νγ^{m}}{sin θ} e^{i( \frac{π}{2} −θz)}) for 0 ⩽ Re z ⩽ 1. Then |φ(z)| = exp(−νγ^{m} \frac{sin θx}{sin θ}e^{θy}), where z = x + iy. We apply the Stein interpolation theorem to the function G defined by

G(z) := φ(z)ρ^{zγ}F (e^{iθ(1−z)})ρ^{−zγ} (0 ⩽ Re z ⩽ 1).

For Re z = 0, the function z ↦ e^{iθ(1−z)} describes the upper ray of the boundary of S_{θ}, for Re z = 1 it describes the positive real semi-axis. For f, g ∈ L_{∞,c}, the function z ↦ ⟨G(z)f, g⟩ is analytic, and we have

|⟨G(z)f, g⟩| ⩽ |φ(z)|||F (e^{iθ(1−z)})|| · ||ρ^{−zγ}f||_{p}||ρ^{zγ}g||_{p′} ⩽ ||F ||_{∞} · c||f||_{p}||g||_{p′} < ∞,

where c depends on γ and on the supports of f and g, but not on z.

The function φ is chosen in such a way that ||G(z)|| ⩽ C_{1} = max{||F ||_{∞}, C} for Re z = 0, 1. We infer that ||G(z)|| ⩽ C_{1} for all 0 ⩽ Re z ⩽ 1, so

||ρ^{xγ}F (e^{iθ(1−x)}e^{θy})ρ^{−xγ}|| ⩽ C_{1}/|φ(x + iy)| = C_{1} exp(νγ^{m} \frac{sin θx}{sin θ}e^{θy}).

Choose now x = 1 − \frac{φ}{θ} and let z := e^{iθ(1−x)}e^{θy} = e^{iφ}e^{θy}. Then

||ρ^{xγ}F (z)ρ^{−xγ}|| ⩽ C_{1} exp(νγ^{m} \frac{sin(θ−φ)}{sin θ} \frac{Re z}{cos φ}).

Writing \frac{γ}{x} = \frac{θ}{θ−φ}γ instead of γ we obtain the assertion with ν_{φ} = ν(\frac{θ}{θ−φ})^{m} \frac{sin(θ−φ)}{sin θ cos φ}.

Now we are in a position to show the following improved version of Theorem 2.1.

Theorem 2.24. Assume that (2.18) holds and that (Ω, d) is separable. Let p_{0} ⩽ s ⩽ q_{0} and T a semigroup on L_{s}(Ω). Assume that there exist ε, t_{0} > 0, m > 1 and α_{0}, β_{0} ⩾ 0 with α_{0} + β_{0} = p_{0}^{−1} − q_{0}^{−1} such that

C_{0} := sup_{0<t⩽t_{0}} ||v_{t^{1/m}}^{α_{0}}T (t)v_{t^{1/m}}^{β_{0}}||_{p_{0}→q_{0},εt^{−1/m}} < ∞.

(a) Then T extrapolates to an exponentially bounded semigroup T_{p} on L_{p}(Ω), for all p ∈ [p_{0}, q_{0}] \ {∞}.

(b) If one of the semigroups T_{p} is strongly continuous then so are all of them.

(c) Assume that T_{p_{1}} has an exponentially bounded analytic extension to S_{θ} for some θ ∈ (0, \frac{π}{2}], p_{1} ∈ [p_{0}, q_{0}] \ \{∞\}. Then T_{p} has an exponentially bounded analytic extension to S_{φ} for all φ ∈ (0, θ), p ∈ [p_{0}, q_{0}] \ {∞}.

(d) In case the assumption of (c) holds let I := [0, θ), otherwise I := {0}, S_{0} := (0, ∞). Then for all φ ∈ I there exist C_{φ} ⩾ 1, ω_{φ} ∈ ℝ and ν_{φ} ⩾ 0 such that

||v_{(Re z)^{1/m}}^{α}T (z)v_{(Re z)^{1/m}}^{β}||_{p→q,γ} ⩽ C_{φ}e^{(ω_{φ}+ν_{φ}|γ|^{m}) Re z}  (2.19)

for all p_{0} ⩽ p ⩽ q ⩽ q_{0}, z ∈ S_{φ}, γ ∈ ℝ, α, β ⩾ 0 with α + β = p^{−1} − q^{−1}. If θ = \frac{π}{2} the one can choose ω_{φ} = ω̃ + (8πc_{1})^{m}(\frac{π}{2} − φ)^{−m} for some ω̃ ∈ ℝ.

Observe that, by our definition of an analytic semigroup, the above theorem implies Theorem 2.1.

Proof of Theorem 2.24. It suffices to show (d): taking α = β = γ = 0, z ∈ (0, ∞), p = q, the weighted norm estimate (2.19) implies (a). If T_{p} is strongly continuous for some p ∈ [p_{0}, q_{0}] \ {∞} then, for q ∈ [p_{0}, q_{0}] \ {1, ∞}, the strong continuity of T_{q} follows from [Voi92]; in the case q = p_{0} = 1 it follows from Lemma 2.20. Finally, part (c) follows from Lemma 1.4.

We first show (2.19) for p = q and φ = 0, i.e., z ∈ (0, ∞). Without restriction assume εt_{0}^{−1/m} ⩾ 8c_{1}. Let t ⩽ t_{0} and γ_{0} := εt^{−1/m}. Note that γ_{0} ⩾ εt_{0}^{−1/m} ⩾ 8c_{1} and γ_{0}t^{1/m} = ε. By Remark 2.21 we obtain

\begin{aligned}||v_{t^{1/m}}^{α}T (t)v_{t^{1/m}}^{β}||_{p→q,γ} & ⩽ K_{ε}e^{5ε}||v_{t^{1/m}}^{α_{0}}T (t)v_{t^{1/m}}^{β_{0}}||_{p_{0}→q_{0},εt^{−1/m}} \\ & ⩽ K_{ε}e^{5ε}C_{0} := C_{1} (t ⩽ t_{0}, γ ⩽ εt^{−1/m}/2) \end{aligned}  (2.20)

for all p_{0} ⩽ p ⩽ q ⩽ q_{0} and α, β ⩾ 0 with α + β = p^{−1} − q^{−1}. By Lemma 2.22 we infer that there exist ω_{1}, ν_{1} ⩾ 0 such that

||T (t)||_{p→p,γ} ⩽ C_{1}e^{ω_{1}t+ν_{1}γ^{m}t} (p ∈ [p_{0}, q_{0}], t, γ ⩾ 0).

In particular, T extrapolates to an exponentially bounded semigroup on L_{p}(Ω) for all p ∈ [p_{0}, q_{0}] \ {∞}.

Assume that T_{p_{1}} has an exponentially bounded extension to S_{θ} for some θ ∈ (0, \frac{π}{2}], p_{1} ∈ [p_{0}, q_{0}] \ \{∞\}. Let C_{2} ⩾ C_{1}, ω_{2} ⩾ ω_{1} such that ||T_{p_{1}}(z)|| ⩽ C_{2}e^{ω_{2} Re z} for all z ∈ S_{θ}. Then Proposition 2.23 implies that for all φ ∈ (0, θ) there exists ν_{2} = ν_{2}(φ) ⩾ 0 such that

||T_{p_{1}}(z)||_{p_{1}→p_{1},γ} ⩽ C_{2}e^{(ω_{2}+ν_{2}γ^{m}) Re z} (z ∈ S_{\frac{φ+θ}{2}} , γ ⩾ 0).  (2.21)

If θ = \frac{π}{2} then we can choose ν_{2}(φ) = (\frac{1}{2} − \frac{φ}{π})^{−m}ν_{1}. If there is no exponentially bounded extension then I = {0}, and (2.21) still holds for φ = 0, with θ := 0.

Now we show (2.19) for p = p_{0}, q = q_{0}, α = α_{1} := p_{1}^{−1} − q_{0}^{−1}, β = β_{1} := p_{0}^{−1} −p_{1}^{−1} and γ > 0. Let φ ∈ I and choose δ > 0 in such a way that z −2δ Re z ∈ S_{\frac{φ+θ}{2}} for all z ∈ S_{φ}. Let z ∈ S_{φ}. Then t_{z} := (δ Re z) ∧ (2γ/ε)^{−m} ∧ t_{0} ⩽ t_{0},

γ ⩽ εt_{z}^{−1/m}/2, and z − 2t_{z} ∈ S_{\frac{φ+θ}{2}} . From (2.20) and (2.21) we therefore obtain, taking into account Remark 2.15,

||v_{t_{z}^{1/m}}^{α_{1}}T_{p_{1}}(z)v_{t_{z}^{1/m}}^{β_{1}}||_{p_{0}→q_{0},γ}

⩽ ||T_{p_{0}}(t_{z})v_{t_{z}^{1/m}}^{β_{1}}||_{p_{0}→p_{1},γ}||T_{p_{1}}(z − 2t_{z})||_{p_{1}→p_{1},γ}||v_{t_{z}^{1/m}}^{α_{1}}T_{p_{1}}(t_{z})||_{p_{1}→q_{0},γ}

⩽ C_{1} · C_{2}e^{(ω_{2}+ν_{2}γ^{m}) Re(z−2t_{z})} · C_{1}.  (2.22)

To obtain the desired estimate (2.19) for v_{r_{z}}^{α_{1}}T_{p_{1}}(z)v_{r_{z}}^{β_{1}}, with r_{z} := (Re z)^{1/m}, we have to estimate (v_{r_{z}}/v_{t_{z}^{1/m}})^{α_{1}+β_{1}}. By (2.18), v_{R}/v_{r} ⩽ c_{0}(1)e^{R/r} for all r ⩽ c_{1}^{−1}, R > r. Without restriction assume t_{0}^{1/m} ⩽ c_{1}^{−1} so that t_{z}^{1/m} ⩽ c_{1}^{−1}. We have t_{z}^{−1/m} = (δ Re z)^{−1/m} ∨ \frac{2γ}{ε} ∨ t_{0}^{−1/m}. Since m > 1 there exists c ⩾ 0 such that

r_{z}/t_{z}^{1/m} ⩽ δ^{−1/m} + (\frac{2γ}{ε} ∨ t_{0}^{−1/m})(Re z)^{1/m} ⩽ c + (1 + γ^{m}) Re z (Re z > 0). (2.23)

Therefore, v_{r_{z}}/v_{t_{z}^{1/m}} ⩽ c_{0}(1)e^{c+(1+γ^{m}) Re z}. Since α_{1}+β_{1} ⩽ 1, estimate (2.22) yields

||v_{r_{z}}^{α_{1}}T_{p_{1}}(z)v_{r_{z}}^{β_{1}}||_{p_{0}→q_{0},γ} ⩽ C_{3}e^{(ω_{3}+ν_{3}γ^{m}) Re z} (z ∈ S_{φ}),

with C_{3} = C_{1}^{2}C_{2}c_{0}(1)e^{c}, ω_{3} = ω_{2} + 1, ν_{3} = ν_{3}(φ) = ν_{2}(φ) + 1.

Finally, let p_{0} ⩽ p ⩽ q ⩽ q_{0}, z ∈ S_{φ}, γ ∈ ℝ and α, β ⩾ 0 with α + β = p^{−1} − q^{−1}. For γ_{0} := (2|γ|) ∨ r_{z}^{−1} ∨ (8c_{1}) we obtain, noting r_{z}^{−m} = (Re z)^{−1}, that

||v_{r_{z}}^{α_{1}}T_{p_{1}}(z)v_{r_{z}}^{β_{1}}||_{p_{0}→q_{0},γ_{0}} ⩽ C_{3}e^{ω_{3} Re z+ν_{3}((2|γ|)^{m}+r_{z}^{−m}+(8c_{1})^{m}) Re z}
= C_{4}e^{(ω_{4}+ν_{4}|γ|^{m}) Re z},

with C_{4} = C_{4}(φ) = C_{3}e^{ν_{3}(φ)}, ω_{4} = ω_{4}(φ) = ω_{3} + (8c_{1})^{m}ν_{3}(φ), ν_{4} = ν_{4}(φ) = 2^{m}ν_{3}(φ). Note that γ_{0} ⩾ \frac{1}{r_{z}} ∨ (8c_{1}) and |γ| ⩽ γ_{0}/2. Thus, Remark 2.21 yields

||v_{r_{z}}^{α}T_{p_{1}}(z)v_{r_{z}}^{β}||_{p→q,γ} ⩽ K_{1}e^{5γ_{0}r_{z}} · C_{4}e^{(ω_{4}+ν_{4}|γ|^{m}) Re z}.

As in (2.23), there exists c ⩾ 1 such that 5γ_{0}r_{z} ⩽ c + (1 + |γ|^{m}) Re z, and we obtain the desired estimate (2.19).

For the case θ = \frac{π}{2} we compute

ω_{φ} := ω_{4}(φ) + 1 = ω_{2} + 2 + (8c_{1})^{m}((\frac{1}{2} − \frac{φ}{π})^{−m}ν_{1} + 1).

This yields the last assertion.

Remark 2.25. Let T (t) = e^{−tA} be a semigroup on a Banach space X which is analytic of angle \frac{π}{2}. Assume that there exist C ⩾ 1, ω ∈ ℝ, ν > 0, m > 1 such that

||T (z)|| ⩽ Ce^{(ω+ν( \frac{π}{2} −arg z)^{−m}) Re z} (Re z > 0).

Then there exist ω_{1} ∈ ℝ, ν_{1} > 0 such that the spectrum σ(A) lies in the ‘filled generalised parabola’

\{x + iy ∈ ℂ; x ⩾ ν_{1}|y|^{\frac{m}{m−1}} − ω_{1}\}.

This easily follows by optimisation from the following fact. If φ ∈ [0, \frac{π}{2}) and ||T (z)|| ⩽ Ce^{ω Re z} for all z ∈ S_{φ} then

σ(A) ⊆ {x + iy ∈ ℂ; x ⩾ |y| tan φ − ω}.

2.5. L_{P} -SPECTRAL INDEPENDENCE 51

2.5 L_{p}-spectral independence

Throughout this section we assume that conditions (2.7) and (2.8) hold. We are going to prove Theorem 2.4 which deals with L_{p}-spectral independence for generators −A_{p} of consistent C_{0}-semigroups T_{p}. By the formula

(λ + A_{p})^{−n} = ∫_{0}^{∞} \frac{t^{n−1}}{(n−1)!}e^{−λt}T_{p}(t) dt,  (2.24)

which holds for λ greater than the type of T_{p}, the theorem will be reduced to the following result on L_{p}-spectral independence for consistent bounded operators.

Theorem 2.26. (cf. [KuVo00; Thms. 1 and 2]) Let B: L_{∞,c} → L_{1,loc} be a linear operator satisfying

||v_{1}^{α}Bv_{1}^{β}||_{p_{0}→q_{0},γ_{0}} < ∞

for some γ_{0} > 0, α, β ⩾ 0 with α+β = p_{0}^{−1}−q_{0}^{−1}. Then B extends to a consistent family of bounded operators B_{p} on L_{p}(Ω), p ∈ [p_{0}, q_{0}]\{∞}, the spectrum σ(B_{p}) is independent of p ∈ [p_{0}, q_{0}]\{∞}, and the operators B_{p} have consistent resolvents.

In [KuVo00] this theorem was proved under the additional assumption that ε ⩽ v_{1} ⩽ ε^{−1} for some ε > 0. The general theorem can be reduced to this case by resorting to the weighted measure space (M, v_{1}^{−1}μ) and using [KuVo00; Thms. 23 and 26] (see [Vog01]). As we are going to present a selfcontained proof here, we will use a direct approach instead.

Using the methods introduced in this section, one can show the following for the case q_{0} = ∞: under the assumptions of Theorem 2.26, B extends to a weak^{∗}-continuous operator B_{∞} on L_{∞}(Ω), and σ(B_{∞}) = σ(B_{p_{0}}). Similarly, the other results of this section hold without the restriction p < ∞, after suitable reformulation for the case p = ∞. We don’t pursue this for the sake of simplicity.

In the case (p_{0}, q_{0}) = (1, ∞), Theorem 2.26 is not optimal in two respects: firstly, a weighted 1→∞-estimate is a strong assumption on the integral kernel of B (cf. Corollary 2.18). It is possible to replace this assumption by an appropriate integrability assumption. Secondly, if we know more than the subexponential volume growth in (2.7), e.g. if the volume growth is polynomial, then the weight functions ρ_{γ,y} can be adjusted to this particular volume growth. See [KuVo00; Thm. 2].

For the proof of Theorem 2.26 we need several preparatory results. The proof of Theorem 2.4 will be given at the end of the section.

In order to prove the inclusion ρ(B_{p}) ⊆ ρ(B_{q}) for p, q ∈ [p_{0}, q_{0}] \ {∞} one has to show that for λ ∈ ρ(B_{p}) the operator (λ − B_{p})^{−1}↾_{L_{∞,c}} extends to a bounded operator on L_{q}(Ω). This is expressed in the following elementary lemma which is stated in the general context of topological spaces (cf. [LiVo00; Prop. 4], [Are94; Prop. 2.3]).

Let E, F, G be Hausdorff spaces with E, F ↪ G (continuous injections) such that E ∩ F is dense in both E and F. Let D ⊆ E ∩ F be a subset which is dense with respect to the initial topology induced by the imbeddings E ∩ F ↪ E and E ∩ F ↪ F.

Lemma 2.27. ([KuVo00; Lemma 9]) Let B_{E}: E → E and B_{F}: F → F be continuous mappings satisfying B_{E}↾_{D} = B_{F} ↾_{D}. Assume that B_{E} is continuously invertible and that B_{E}^{−1}↾_{D} extends to a continuous mapping R: F → F. Then B_{F} is continuously invertible, and B_{F}^{−1} = R.

Proof. Since D is dense in E ∩ F and E, F ↪ G, we have B_{E}↾_{E∩F} = B_{F} ↾_{E∩F} and B_{E}^{−1}↾_{E∩F} = R↾_{E∩F}. Hence RB_{F} = B_{F} R = I on E ∩ F. This yields the claim since E ∩ F is dense in F.

We will apply this lemma in the situation E = L_{p}(Ω), F = L_{q}(Ω), G = L_{1,loc} and D = L_{∞,c}.

In the proof of Theorem 2.26, we will make use of Proposition 2.16 in the following form.

Remark 2.28. Let B: L_{∞,c} → L_{1,loc} be a linear operator, γ_{0} > 0. Then

||v_{1}^{α}Bv_{1}^{β}||_{p→q,γ} ⩽ K_{γ_{0}}||v_{1}^{α_{0}}Bv_{1}^{β_{0}}||_{p_{0}→q_{0},γ_{0}}

for all p_{0} ⩽ p ⩽ q ⩽ q_{0}, |γ| ⩽ γ_{0}/2, and α_{0}, β_{0}, α, β ⩾ 0 with α_{0} + β_{0} = p_{0}^{−1} − q_{0}^{−1}, α + β = p^{−1} − q^{−1}. Here we can choose K_{γ_{0}} := c_{γ_{0}/8}^{6}(1 − e^{−γ_{0}/8})^{−2}e^{5γ_{0}/8}, with c_{γ_{0}/8} from (2.7).

For the proof, apply Proposition 2.16 with c_{r} := \frac{γ_{0}}{8} and c_{0} := c_{γ_{0}/8}.

The crucial part in the proof of Theorem 2.26 is the following estimate which implies convergence of weighted operators (cf. [Sch96; Lemma 3.2.3], [LiVo00; Prop. 5(iii)]).

Proposition 2.29. (cf. [KuVo00; Prop. 15]) Let B: L_{∞,c} → L_{1,loc} be a linear operator, 1 ⩽ p < ∞, γ_{0} > 0. There exist δ_{γ_{0},γ} > 0 with δ_{γ_{0},γ} → 0 as γ → 0 such that

||ρBρ^{−1} − B||_{p→p} ⩽ δ_{γ_{0},γ}||B||_{p→p,γ_{0}}

for all 0 < γ < γ_{0} and ρ ∈ P (γ, 1). In particular, for all |γ| < γ_{0}, ρ_{γ,y}Bρ_{γ,y}^{−1} extends to a bounded operator B_{γ,y} on L_{p}(Ω), and B_{γ,y} → B_{0,y} in the norm as γ → 0, uniformly in y ∈ M.

Proof. Let 0 < γ < γ_{0} and ρ ∈ P (γ, 1). By (2.7) there exists c_{0} > 0 such that v_{r} ⩽ c_{0}e^{(γ_{0}−γ)r}v_{1} for all r > 1. By (2.14) this implies v := v_{1}^{−1/p} ∈ P (γ_{0} − γ, C), with C := c_{0}e^{γ_{0}−γ}.

Let f ∈ L_{∞,c}. By Lemma 2.19 we have

||(ρBρ^{−1} − B)f||_{p} ⩽ e^{2γ_{0}} ||y ↦ ||ρ_{2γ_{0},y}v(ρBρ^{−1} − B)f||_{p}||_{L_{p}(M)}.  (2.25)

We now write

ρBρ^{−1} − B = ρρ(y)^{−1}B(ρ(y)ρ^{−1} − 1) + (ρρ(y)^{−1} − 1)B =: B_{1} + B_{2},

insert this into (2.25), use the triangle inequality and estimate the two resulting terms separately. For the second term we have, using Lemma 2.13(b) and (2.15),

||y ↦ ||ρ_{2γ_{0},y}vB_{2}f||_{p}||_{L_{p}(M)} ⩽ n_{p}(v(x)ρ_{2γ_{0},y}(x)(ρ_{−γ,y}(x) − 1))||B||_{p→p}||f||_{p}.

Using (2.15) and Lemma 2.13(b) again, we can estimate the first term by

||y ↦ ||ρ_{2γ_{0},y}vB_{1}f||_{p}||_{L_{p}(M)} ⩽ C||y ↦ ||ρ_{γ_{0},y}v(y)B(ρ(y)ρ^{−1} − 1)f||_{p}||_{L_{p}(M)}

⩽ C||B||_{p→p,γ_{0}} ||y ↦ ||ρ_{γ_{0},y}v(y)(ρ_{−γ,y} − 1)f||_{p}||_{L_{p}(M)}

⩽ C||B||_{p→p,γ_{0}}n_{p}(v(y)ρ_{γ_{0},y}(x)(ρ_{−γ,y}(x) − 1))||f||_{p}.

To complete the proof, note that ||B||_{p→p} ⩽ K_{γ_{0}}||B||_{p→p,γ_{0}} by Remark 2.28 and that

n_{p}(v(x)ρ_{γ_{0},y}(x)(ρ_{−γ,y}(x) − 1)) ⩽ n_{p}(v(x)e^{−γ_{0}d(x,y)/2}) sup_{r⩾0}e^{−γ_{0}r/2}(e^{γr} − 1) → 0

as γ → 0, where we used ρ_{γ_{0},y} = (ρ_{γ_{0}/2,y})^{2}.

The following consequence of Proposition 2.29 will be used in the proof of Theorem 2.26.

Corollary 2.30. (cf. [KuVo00; Cor. 16]) Let 1 ⩽ p < ∞ and B a bounded operator on L_{p}(Ω) satisfying ||B||_{p→p,γ_{0}} < ∞ for some γ_{0} > 0. Let λ ∈ ρ(B). Then there exists γ_{1} > 0 such that ||(λ − B)^{−1}||_{p→p,γ} < ∞ for all |γ| ⩽ γ_{1}.

Proof. Let λ ∈ ρ(B). By the assumption we have ||λ − B||_{p→p,γ_{0}} < ∞. Recall that inversion is continuous in the open set of invertible elements in L(L_{p}(Ω)). Thus, by Proposition 2.29 there exists γ_{1} > 0 such that λ − B_{γ,y} is invertible for all y ∈ M, |γ| < γ_{1}, and

sup_{y∈M} ||(λ − B_{γ,y})^{−1}||_{p→p} < ∞.

In order to prove ||(λ−B)^{−1}||_{p→p,γ} < ∞ it remains to show (λ−B_{γ,y})^{−1}↾_{D(ρ_{γ,y}^{−1},L_{p})} = ρ_{γ,y}(λ − B)^{−1}ρ_{γ,y}^{−1} for all y ∈ M. The latter is a consequence of the following two facts: (λ − B_{γ,y})↾_{D(ρ_{γ,y}^{−1},L_{p})} = ρ_{γ,y}(λ − B)ρ_{γ,y}^{−1} for all y ∈ M, and the operator ρ_{γ,y}^{−1}: D(ρ_{γ,y}^{−1}, L_{p}) → L_{p}(Ω) is bijective since ρ_{γ,y} is bounded.

Proof of Theorem 2.26. By Remark 2.28 it is clear that B extends to a bounded operator B_{p} on L_{p}(Ω), for all p ∈ [p_{0}, q_{0}]\{∞}. Let p, q ∈ [p_{0}, q_{0}]\{∞}. The operators B_{p} and B_{q} are consistent since L_{∞,c} is dense in L_{p}(Ω) ∩ L_{q}(Ω). So we have to prove the inclusion ρ(B_{p}) ⊆ ρ(B_{q}) and the consistency of the resolvents. Let λ ∈ ρ(B_{p}).

First we study the case λ ≠ 0. Then we can rewrite the resolvent R(λ) of B_{p} as

R(λ) = λ^{−1}I + λ^{−2}B_{p} + λ^{−2}B_{p}R(λ)B_{p}.

We have to show that λ ∈ ρ(B_{q}) and that (λ − B_{p})^{−1}, (λ − B_{q})^{−1} are consistent, which by Lemma 2.27 amounts to showing L_{q}-boundedness of R(λ)↾_{L_{∞,c}}.

It is clear that λ^{−1}I + λ^{−2}B is L_{q}-bounded; we will show that B_{p}R(λ)B is L_{q}- bounded. According to Corollary 2.30 we have ||R(λ)||_{p→p,γ} < ∞ for some 0 < γ < \frac{γ_{0}}{2} . Let α := p_{0}^{−1}−p^{−1}, β := p^{−1}−q_{0}^{−1}. Then ||Bv_{1}^{α}||_{p_{0}→p,γ} < ∞ and ||v_{1}^{β}B||_{p→q_{0},γ} < ∞ by Remark 2.28. Remark 2.15 implies that ||v_{1}^{β}B_{p}R(λ)Bv_{1}^{α}||_{p_{0}→q_{0},γ} < ∞. Another application of Remark 2.28 yields L_{q}-boundedness of B_{p}R(λ)B.

In the case λ = 0 we simply write R(λ) = B_{p}R(λ)^{3}B_{p}. By the above we have ||v_{1}^{β}R(λ)v_{1}^{α}||_{p_{0}→q_{0},γ} < ∞. Again ||R(λ)||_{q→q} < ∞ by Remark 2.28.

Remark 2.31. Let 1 ⩽ p < q ⩽ ∞ and α := p^{−1} − q^{−1}. Let B_{p} be a bounded operator on L_{p}(Ω) with 0 ∈ ρ(B_{p}) and ||v_{1}^{α}B_{p}||_{p→q} < ∞. Then ||v_{1}^{α}: L_{p}(Ω) → L_{q}(Ω)|| ⩽ ||B_{p}^{−1}||_{p→p}||v_{1}^{α}B_{p}||_{p→q} < ∞. Therefore (Ω, μ) cannot contain a sequence (M_{n}) of bounded subsets satisfying M_{n} ⊇ M_{n+1} (n ∈ ℕ) and 0 < μ(M_{n}) → 0 (n → ∞).

Then Ω = ⋃_{n=0}^{∞} M_{n} where μ(M_{0}) = 0, and M_{n} are pairwise disjoint atoms of (Ω, μ) (n ⩾ 1). Therefore, for all s < ∞, the space L_{s}(Ω) is isometrically isomorphic to the weighted space of sequences {(x_{n}); ∑_{n} |x_{n}|^{s}μ(M_{n}) < ∞}, and L_{∞}(Ω) is isometrically isomorphic to l_{∞}. In this case we have ||B_{p}↾_{L_{1}(Ω)∩L_{p}(Ω)}||_{1→∞} < ∞.

In order to derive Theorem 2.4 from Theorem 2.26 we need some more preparation. Part (b) of the following lemma is inspired by [Kar00; Lemma 6.3].

Lemma 2.32. Let A be a closed operator in a Banach space X, and n ∈ ℕ.

(a) Let λ_{0} ∈ ρ(A), λ ∈ ℂ such that (λ_{0} − λ)^{−n} ∈ ρ((λ_{0} − A)^{−n}). Then λ ∈ ρ(A), and

(λ − A)^{−1} = ∑_{k=1}^{n}(λ_{0} − λ)^{k−n−1}(λ_{0} − A)^{−k}((λ_{0} − λ)^{−n} − (λ_{0} − A)^{−n})^{−1}.

(b) Assume that σ(A) ⊆ {λ ∈ ℂ; Re λ ⩽ ω} for some ω ∈ ℝ. Then for all λ ∈ ρ(A) there exists λ_{0} > ω such that (λ_{0} − λ)^{−n} ∈ ρ((λ_{0} − A)^{−n}).

Proof. (a) Let B := (λ_{0}−A)^{−1}, α := (λ_{0}−λ)^{−1}, S := ∑_{1}^{n}(λ_{0}−λ)^{k−n−1}(λ_{0}−A)^{−k}. Then

(λ − A)S = (B^{−1} − α^{−1}) ∑_{k=1}^{n} α^{n+1−k}B^{k} = (α − B) ∑_{k=1}^{n} α^{n−k}B^{k−1} = α^{n} − B^{n},

and hence (λ − A)S(α^{n} − B^{n})^{−1} = id. In the same way, S(α^{n} − B^{n})^{−1}(λ − A) = id_{D(A)} (note that S and (α^{n} − B^{n})^{−1} commute). This proves (a).

(b) Let λ ∈ ρ(A). By the spectral mapping theorem for bounded operators we have to show that there exists λ_{0} > ω such that

(λ_{0} − λ)^{−1}e^{2πi \frac{k}{n}} ∈ ρ((λ_{0} − A)^{−1}) (k = 0, …, n − 1).

By the spectral mapping theorem for the resolvent, the latter is equivalent to μ_{k} := λ_{0} − (λ_{0} − λ)e^{2πi \frac{k}{n}} ∈ ρ(A) (k = 0, …, n − 1). For k = 0 this is true since λ ∈ ρ(A). For k = 1, …, n − 1 we have Re μ_{k} ⩾ λ_{0}(1 − cos \frac{2πk}{n}) − |λ|, so by the assumption on σ(A) we obtain that μ_{k} ∈ ρ(A) if λ_{0} is sufficiently large.

For the next result let E, F be Banach spaces, and assume that there exists a Hausdorff topological vector space G such that E ↪ G, F ↪ G (continuous injections) and E ∩ F is dense in both E and F.

Proposition 2.33. Let A_{E}, A_{F} be closed operators in E, F, respectively, with σ(A_{E}), σ(A_{F}) ⊆ {λ ∈ ℂ; Re λ ⩽ ω} for some ω ∈ ℝ. Let n ∈ ℕ. Assume that, for all λ_{0} > ω, the resolvents (λ_{0} − A_{E})^{−1}, (λ_{0} − A_{F})^{−1} are consistent, and (λ_{0} −A_{E})^{−n}, (λ_{0} −A_{F})^{−n} have equal spectra and consistent resolvents. Then A_{E}, A_{F} have equal spectra and consistent resolvents.

Proof. We have to show ρ(A_{E}) ⊆ ρ(A_{F}) and the consistency of the resolvents. Let λ ∈ ρ(A_{E}). By Lemma 2.32(b) there exists λ_{0} > ω such that (λ_{0} − λ)^{−n} ∈ ρ((λ_{0} − A_{E})^{−n}), so (λ_{0} − λ)^{−n} ∈ ρ((λ_{0} − A_{F})^{−n}) by the assumption. Since λ_{0} ∈ ρ(A_{E}) ∩ ρ(A_{F}), the assertions of Lemma 2.32(a) are fulfilled for A = A_{E} as well as A = A_{F}. Thus, λ ∈ ρ(A_{F}), and (λ − A_{E})^{−1}, (λ − A_{F})^{−1} are consistent since (λ_{0} − A_{E})^{−1}, (λ_{0} − A_{F})^{−1} are consistent and (λ_{0} − A_{E})^{−n}, (λ_{0} − A_{F})^{−n} have consistent resolvents.

Proof of Theorem 2.4. We are going to show a weighted norm estimate for (λ + A_{p})^{−n} in order to apply Theorem 2.26 to this operator.

By Remark 2.28 it follows from assumption (2.9) that

||v_{1}^{p_{0}^{−1}−p^{−1}}T_{p_{0}}(t)||_{p_{0}→p,γ_{0}/2} ⩽ K_{γ_{0}} · C(\frac{t_{0}}{2} )^{−K} =: C_{0} (\frac{t_{0}}{2} ⩽ t ⩽ t_{0}, p_{0} ⩽ p ⩽ q_{0}).

With α := p_{0}^{−1} − q_{0}^{−1} and ω := t_{0}^{−1} ln C_{0} we deduce that

||v_{1}^{α}T_{p_{0}}(t)||_{p_{0}→q_{0},γ_{0}/2} ⩽ C_{0}e^{ωt} (t ⩾ t_{0}).

For small times, Remark 2.28 applied to assumption (2.9) yields

||v_{1}^{α}T_{p_{0}}(t)||_{p_{0}→q_{0},γ_{0}/2} ⩽ K_{γ_{0}} · Ct^{−K} (0 < t ⩽ t_{0}).

Let n ∈ ℕ with n > K. For λ_{0} > ω we obtain, using the representation of (λ_{0} + A_{p_{0}})^{−n} given in (2.24),

||v_{1}^{α}(λ_{0} + A_{p_{0}})^{−n}||_{p_{0}→q_{0},γ_{0}/2} < ∞.

Note that the operators (λ_{0} + A_{p})^{−n} are consistent, by (2.24). Therefore, by Theorem 2.26, they have spectrum independent of p ∈ [p_{0}, q_{0}]\{∞} and consistent resolvents. We conclude the proof by an application of Proposition 2.33.

In the case (p_{0}, q_{0}) ≠ (1, ∞) we could avoid using (2.24), and we would need Proposition 2.33 only for the simple case n = 1: By the assumption of Theorem 2.4, for all r ∈ [p_{0}, q_{0}] \ {∞} there exists C_{r} ⩾ C such that ||T_{r}(t)|| ⩽ C_{r} (t ⩽ t_{0}). Let p, q ∈ [p_{0}, q_{0}] \ {∞} with p < q, and let θ := θ_{p,q} ∈ (0, 1] such that p^{−1} −q^{−1} = θ(p_{0}^{−1} −q_{0}^{−1}). Then Stein interpolation between the bounds (2.9) and ||T_{r}(t)|| ⩽ C_{0} for suitable r ∈ [p, q] yields

||v_{1}^{θα}T_{p}(t)v_{1}^{θβ}||_{p→q,θγ_{0}} ⩽ C_{r}t^{−θK} (0 < t ⩽ t_{0}).

Now assume that θ_{p,q} < \frac{1}{K}, i.e., p^{−1} − q^{−1} < \frac{1}{K} (p_{0}^{−1} − q_{0}^{−1}). Then we obtain, as in the above proof,

||v_{1}^{θα}(λ_{0} + A_{p})^{−1}v_{1}^{θβ}||_{p→q,θγ_{0}} < ∞.  (2.26)

It remains to apply Theorem 2.26 and the spectral mapping theorem for the resolvent to obtain that σ(A_{r}) is independent of r ∈ [p, q], whenever p^{−1} − q^{−1} < \frac{1}{K} (p_{0}^{−1} − q_{0}^{−1}). The consistency of the resolvents of the operators A_{r} is straightforward from the consistency of the resolvents of the operators (λ_{0} + A_{r})^{−1}. This proves Theorem 2.4.

Observe that the above idea of proof is not applicable in the case (p_{0}, q_{0}) = (1, ∞): then we only have the assumption v_{r}(x) ⩽ μ(B(x, r)). Hence the weighted p→q-estimate (2.26) is of no use —Theorem 2.26 requires a weighted 1→∞- estimate.

L_{p}-properties of elliptic differential operators

This chapter is devoted to the L_{p}-theory of second order elliptic differential operators on an open set Ω ⊆ ℝ^{N}, N ∈ ℕ, corresponding to the formal differential expression

L := −∇ · (a∇) + b_{1} · ∇ + ∇ · b_{2} + V

with singular measurable coefficients a: Ω → ℝ^{N} ⊗ ℝ^{N}, b_{1}, b_{2}: Ω → ℝ^{N}, V: Ω → ℝ. We are going to construct a positive C_{0}-semigroup on L_{p} := L_{p}(Ω), whose generator is associated with L in a natural way which will be made precise below. As it is well-known, this implies well-posedness of the corresponding Cauchy problem.

There is vast literature concerning the case that one can associate a consistent family of C_{0}-semigroups on all L_{p}-spaces with the differential expression L. This, however, is not the case of major interest here. In general, L will be associated with a C_{0}-semigroup on L_{p} for p from a proper subinterval of [1, ∞).

The chapter is organised as follows. In Section 3.1 we motivate and formulate our main results concerning the construction of a family of positive C_{0}-semigroups on L_{p} associated with L. Moreover, we investigate the problems of extrapolation, analyticity and L_{p}-spectral independence for these semigroups. The proofs of the main theorems are given in the two subsequent sections. Finally, in Section 3.4 we discuss to what extent our results are sharp.

The contents of this chapter are partly contained in [SoVo00], [LSV00].

3.1 Construction of the semigroup on L_{p} and main properties

Elliptic operators in divergence form with measurable coefficients are usually defined by means of the form method. The form associated with the above differential expression L is defined by

τ(u, v) := ⟨a∇u, ∇v⟩ + ⟨∇u, b_{1}v⟩ − ⟨b_{2}u, ∇v⟩ + ⟨V u, v⟩  (3.1)

on a suitable domain D(τ) corresponding to the boundary conditions. Here and in the following, ⟨f, g⟩ is defined as ∫_{Ω} f(x) · \overline{g}(x) dx whenever f · \overline{g} ∈ L_{1}, for f, g: Ω → ℂ or f, g: Ω → ℂ^{N} measurable. We will consider the lower order terms of τ as (not necessarily form bounded) perturbations of the second order term.

Assume that τ is densely defined and fulfils the first Beurling-Deny criterion. (The latter holds if and only if (Re u)^{+} ∈ D(τ) for all u ∈ D(τ); in this case we have τ(u, v) ∈ ℝ and τ(u^{+}, u^{−}) = 0 for all real-valued u, v ∈ D(τ)—see Definition 1.12.) Then the precise formulation of the problem is as follows: Given p ∈ [1, ∞), under which conditions on the coefficients a, b_{1}, b_{2}, V and the domain D(τ) is τ associated with a positive C_{0}-semigroup T_{p} on L_{p}, in the sense of Definition 1.20? If τ is associated with T_{p}(t) = e^{−tA_{p}}, we can regard A_{p} as the L_{p}-realisation of L with boundary conditions prescribed by D(τ).

The present section is organised as follows. First we formulate conditions on the form τ ensuring that τ is densely defined and fulfils the first Beurling-Deny criterion. We then investigate for which p ∈ [1, ∞) one can expect τ to be associated with a quasi-contractive C_{0}-semigroup on L_{p}. A comprehensive answer is given in Theorem 3.2; see also Corollary 3.5.

In general, the set of all p ∈ [1, ∞) such that τ is associated with a positive C_{0}- semigroup T_{p} on L_{p} is strictly larger than the set I determined in Theorem 3.2. In Theorem 3.8 we will show, under some additional restrictions, that I can be extended to the left and to the right if a is uniformly elliptic. Moreover, we show that under these restrictions we have p-independence of the angle of analyticity and of the L_{p}-spectrum. Our last result, Theorem 3.10, shows that the conditions needed to obtain L_{p}-spectral independence, for p ∈ I only, are considerably weaker.

We make the following qualitative assumptions on the coefficients of L.

(a) a is a.e. invertible, a, a^{−1} ∈ L_{1,loc}, and a is sectorial, i.e.,

|Im(aζ · \overline{ζ})| ⩽ α Re(aζ · \overline{ζ}) a.e. for all ζ ∈ ℂ^{N}

for some α ⩾ 0. Recall from Proposition 1.28 that

τ_{N}(u, v) := ⟨a∇u, ∇v⟩, D(τ_{N}) := \{u ∈ W_{1,loc}^{1} ∩ L_{2}; a∇u · ∇\overline{u} ∈ L_{1}\}

defines a (non-symmetric) Dirichlet form in L_{2}. Let τ_{a} ⊆ τ_{N} be a Dirichlet form.

(bV) The potentials W_{j} := b_{j}^{⊤}a_{s}^{−1}b_{j} (j = 1, 2) and V^{+} are τ_{a}-regular, and Q(V^{−}) ⊇ D(τ_{a}) ∩ Q(W_{1} + W_{2} + V^{+}) (recall that a_{s} = \frac{a+a^{⊤}}{2}).

We define the form τ on D(τ) := D(τ_{a}) ∩ Q(W_{1} + W_{2} + V^{+}) by (3.1). This is possible since for all u, v ∈ D(τ) and j = 1, 2 we have, by the Cauchy-Schwarz inequality,

|∇u · b_{j}\overline{v}| = |a_{s}^{1/2}∇u · a_{s}^{−1/2}b_{j}\overline{v}| ⩽ (a_{s}∇u · ∇\overline{u})^{1/2}(W_{j}|v|^{2})^{1/2} ∈ L_{1}.  (3.2)

Since τ_{a} is a Dirichlet form we have (Re u)^{+} ∈ D(τ) for all u ∈ D(τ). Therefore, τ fulfils the first Beurling-Deny criterion. Of course, we can define τ in the same way as above without assuming (bV). The reason for assuming (bV) is that then D(τ) is dense in D(τ_{a}), by Lemma 1.24(b); in particular, τ is densely defined.

Our first aim is to determine the interval of those p ∈ [1, ∞) for which τ is associated with a quasi-contractive C_{0}-semigroup on L_{p}. The only quantitative condition we need is seen from the Lumer-Phillips theorem by a formal computation. Suppose τ is associated with a positive quasi-contractive C_{0}-semigroup T_{p}(t) = e^{−tA_{p}} on L_{p}, for some 1 < p < ∞. Then A_{p} is quasi-accretive which, by the positivity of T_{p}, is equivalent to ⟨A_{p}u, u^{p−1}⟩ ⩾ −ω_{p}||u||_{p}^{p} for all 0 ⩽ u ∈ D(A_{p}), for some ω_{p} ∈ ℝ.

Formally, A_{p}u = Lu, ∇u^{p−1} = (p − 1)u^{p−2}∇u = \frac{2}{p′} u^{\frac{p}{2} −1}∇u^{\frac{p}{2}} , and similarly ∇u = \frac{2}{p}u^{1− \frac{p}{2}} ∇u^{\frac{p}{2}} . Thus,

⟨A_{p}u, u^{p−1}⟩ = ⟨−∇ · (a∇u) + b_{1} · ∇u + ∇ · (b_{2}u) + V u, u^{p−1}⟩
= \frac{4}{pp′} ⟨a∇u^{\frac{p}{2}} , ∇u^{\frac{p}{2}} ⟩ + ⟨(\frac{2}{p}b_{1} − \frac{2}{p′} b_{2})u^{\frac{p}{2}} , ∇u^{\frac{p}{2}} ⟩ + ⟨V u^{p}⟩

(here and in the following, ⟨f⟩ := ∫_{Ω} f(x) dx for all f ∈ L_{1}). We define symmetric forms τ_{p} on D(τ_{p}) := D(τ) (1 < p < ∞), as real parts of sesquilinear forms, by

τ_{p}(u) := Re(\frac{4}{pp′} ⟨a∇u, ∇u⟩ + \frac{2}{p}⟨∇u, b_{1}u⟩ − \frac{2}{p′} ⟨b_{2}u, ∇u⟩ + ⟨V u, u⟩)
= \frac{4}{pp′} Re τ_{a}(u) + \frac{2}{p}⟨∇|u|, b_{1}|u|⟩ − \frac{2}{p′} ⟨b_{2}|u|, ∇|u|⟩ + ⟨V |u|^{2}⟩

(note that Re(\overline{u}∇u) = |u|∇|u| for all u ∈ W_{1,loc}^{1} [LeSi81; Appendix, Cor. 1]). Then the natural condition for L_{p}-accretivity is

τ_{p}(u) ⩾ −ω_{p}||u||_{2}^{2} for all 0 ⩽ u ∈ D(τ).

This is equivalent to τ_{p} ⩾ −ω_{p} since τ_{p} fulfils the first Beurling-Deny criterion. Note that τ_{2} = Re τ (as to be expected). Further, we define the symmetric forms τ_{1}, τ_{∞} by setting p = 1, ∞ in the above definition:

τ_{1}(u) := 2⟨∇|u|, b_{1}|u|⟩ + ⟨V |u|^{2}⟩,
τ_{∞}(u) := −2⟨b_{2}|u|, ∇|u|⟩ + ⟨V |u|^{2}⟩

on D(τ_{1}) := D(τ_{∞}) := D(τ).

Now we can explain why we do not just assume V^{−} to be τ_{a}-regular in (bV). Suppose that τ_{p} ⩾ −ω_{p} for some p ∈ [1, ∞], ω_{p} ∈ ℝ. Then

⟨V^{−}u^{2}⟩ ⩽ \frac{4}{pp′} Re τ_{a}(u) + ⟨(\frac{2}{p}b_{1} − \frac{2}{p′} b_{2})u, ∇u⟩ + ⟨(V^{+} + ω_{p})u^{2}⟩ (0 ⩽ u ∈ D(τ)).

If V^{−} is τ_{a}-regular then it is (τ_{a} + W_{1} + W_{2} + V^{+})-regular, by Lemma 1.24(b). Thus, the above inequality is valid for all 0 ⩽ u ∈ D(τ_{a}) ∩ Q(W_{1} + W_{2} + V^{+}), so (bV) holds. In other words, if the assumption on V^{−} in (bV) does not hold then none of the forms τ_{p} is bounded below.

The forms τ_{p} play a crucial role in all our results on elliptic operators. In the following proposition we collect several simple properties of the forms τ and τ_{p} which are important for the understanding of the subsequent theorems.

Proposition 3.1. Assume that (a) and (bV) hold. Let I be the set of all p ∈ [1, ∞) such that ω_{p} := inf{ω ∈ ℝ; τ_{p} ⩾ −ω} < ∞ (then τ_{p} ⩾ −ω_{p} for all p ∈ I).

(a) For all potentials U ⩾ W_{1} + W_{2} + V^{−}, the form τ + U is sectorial and closed. For all 1 < p < ∞ and U ⩾ p′W_{1} +pW_{2} +V^{−}, the symmetric form τ_{p} +U is closed. In particular, τ_{p} is closable for all p ∈ I \ {1}.

(b) The set I is an interval and, for all p ∈ ^{∘} I, there exist ε_{p} > 0, c_{p} ∈ ℝ such that τ_{p} ⩾ ε_{p} Re τ_{a} − c_{p}. If τ_{p_{j}} ⩾ −ω_{p_{j}} (j = 0, 1) for some 1 ⩽ p_{0} < p < p_{1} ⩽ ∞ then we can choose ε_{p} = 4(\frac{1}{p_{0}} − \frac{1}{p})(\frac{1}{p} − \frac{1}{p_{1}} ), c_{p} = ω_{p_{0}} ∨ ω_{p_{1}}.

(c) For all p, q ∈ ^{∘} I, the norms || · ||_{τ_{p}} and || · ||_{τ_{q}} are equivalent.

Note that, in case τ_{1}, τ_{∞} ⩾ 0, part (b) of the proposition reads τ_{p} ⩾ \frac{4}{pp′} Re τ_{a}.

Proof of Proposition 3.1. (a) From (3.2) we deduce by Euclid’s inequality (|ab| ⩽ \frac{ε}{2}a^{2} + \frac{1}{2ε}b^{2} for all a, b ∈ ℝ, ε > 0) that the sum of the first order terms of τ is form small with respect to τ_{a} + W_{1} + W_{2}. Thus, τ + U is a closed sectorial form for any potential U ⩾ W_{1} + W_{2} + V^{−}. The analogous argument works for τ_{p} if 1 < p < ∞. By Corollary 1.16 we obtain that τ_{p} is closable if it is bounded below, i.e., p ∈ I.

The proof of (b) and (c) relies on the following identity which results directly from the definition of the forms τ_{p}: for all p_{0}, p_{1} ∈ I, θ ∈ (0, 1) and p_{θ} defined by \frac{1}{p_{θ}} = \frac{1−θ}{p_{0}} + \frac{θ}{p_{1}} we have

τ_{p_{θ}} = (1 − θ)τ_{p_{0}} + θτ_{p_{1}} + 4 (\frac{1}{p_{θ}p′_{θ}} − \frac{1 − θ}{p_{0}p′_{0}} − \frac{θ}{p_{1}p′_{1}} ) Re τ_{a}.  (3.3)

In order to prove (b), it now suffices to show that

\frac{1}{p_{θ}p′_{θ}} − \frac{1 − θ}{p_{0}p′_{0}} − \frac{θ}{p_{1}p′_{1}} = (\frac{1}{p′_{θ}} − \frac{1}{p′_{0}} ) (\frac{1}{p_{θ}} − \frac{1}{p_{1}} ) (= (\frac{1}{p_{0}} − \frac{1}{p_{θ}} ) (\frac{1}{p_{θ}} − \frac{1}{p_{1}} )),

which in turn follows from the equality

\frac{1}{p_{θ}p′_{0}} + \frac{1}{p′_{θ}p_{1}} = (\frac{1 − θ}{p_{0}} + \frac{θ}{p_{1}} ) \frac{1}{p′_{0}} + (\frac{1 − θ}{p′_{0}} + \frac{θ}{p′_{1}} ) \frac{1}{p_{1}} = \frac{1 − θ}{p_{0}p′_{0}} + \frac{θ}{p_{1}p′_{1}} + \frac{1}{p′_{0}p_{1}} .

(c) By (3.3) we have τ_{p_{θ}} ⩾ (1 − θ)τ_{p_{0}} + θτ_{p_{1}}. We deduce that, for all p, q ∈ ^{∘} I, there exist ε > 0, ω ∈ ℝ such that τ_{p} ⩾ ετ_{q} − ω and τ_{q} ⩾ ετ_{p} − ω.

The form τ itself need not be sectorial. In fact, Theorem 3.2 below includes cases where τ is not even bounded from the left. However, the form τ + W_{1} + W_{2} + V^{−} is sectorial and closed by Proposition 3.1(a). This enables us to make use of Definition 1.20 in the first main theorem of this chapter which reads as follows.

Theorem 3.2. Assume that (a) and (bV) hold. Let I be the interval of all p ∈ [1, ∞) such that ω_{p} = inf{ω ∈ ℝ; τ_{p} ⩾ −ω} < ∞. Then the following assertions hold.

(a) The form τ is associated with a consistent family of positive C_{0}-semigroups T_{p}(t) = e^{−tA_{p}} on L_{p}, p ∈ I, with ||T_{p}(t)|| ⩽ e^{ω_{p}t} for all p ∈ I, t ⩾ 0.

(b) For all p ∈ I \ \{1\} and u ∈ D(A_{p}) we have |u|^{\frac{p}{2}} sgn u ∈ D(\overline{τ_{p}}) and

Re⟨A_{p}u, u|u|^{p−2}⟩ ⩾ \overline{τ_{p}}(|u|^{\frac{p}{2}} sgn u).  (3.4)

(c) If, in addition,

|Im⟨(b_{1} + b_{2})u, ∇u⟩| ⩽ c_{1}τ_{p}(u) + c_{2}||u||_{2}^{2} (u ∈ D(τ))  (3.5)

for some p ∈ ^{∘} I, c_{1} ⩾ 0, c_{2} ∈ ℝ then A_{p} is an m-sectorial operator for all p ∈ ^{∘} I, in particular, T_{p} extends to an analytic semigroup on L_{p}.

Remarks 3.3. (a) We point out that the semigroups T_{p} are associated with the form τ, not with the forms τ_{p}. These forms, however, determine important properties of the semigroups T_{p}.

The method used in the theorem to construct the semigroups T_{p} is much more natural than it may seem at first sight. The construction amounts to an approximation of the lower order perturbations: not of the first order terms as one might expect but of the potential—recall that we apply Definition 1.20. See also Corollary 3.4(b) below.

(b) The domain of τ_{a} determines the ‘boundary conditions’ under consideration (the standard examples are the case of Neumann boundary conditions τ_{a} = τ_{N} and of Dirichlet boundary conditions τ_{a} = τ_{D} := \overline{τ_{N}↾_{C_{c}^{∞}(Ω)}}). Assumption (bV) expresses that the lower order perturbations must not disturb the boundary conditions prescribed by D(τ_{a}). In the case of Dirichlet boundary conditions, assumption (bV) is fulfilled in particular if W_{1}, W_{2}, V ∈ L_{1,loc}.

Suppose that assumption (bV) is not fulfilled, but D(τ) is dense in L_{2}. Let τ̃_{a} := \overline{τ_{N}↾_{D(τ)}}. Then assumptions (a) and (bV) are fulfilled with τ̃_{a} in place of τ_{a}, so Theorem 3.2 is still applicable to the form τ. (Note that τ̃_{a} is a Dirichlet form since condition (1.4) is fulfilled for D = D(τ)—see Definition 1.26.)

(c) If the form τ itself is sectorial then it is closable by Proposition 3.1(a) and Corollary 1.16. In this case we have 2 ∈ I, and the operator A_{2} constructed in Theorem 3.2 is just the m-sectorial operator associated with \overline{τ} (cf. the paragraph following Definition 1.20).

(d) We point out that the case I = {1} is quite possible. By definition, 1 ∈ I if τ_{1} ⩾ −ω for some ω ∈ ℝ. Note that the coefficient b_{2} is not involved in this condition. In particular, if (a) holds, b_{1} = 0 and V ⩾ 0 then τ is associated with a positive contractive C_{0}-semigroup on L_{1}, whenever b_{2}^{⊤}a_{s}^{−1}b_{2} is τ_{a}-regular.

(e) For the case p = ∞ we obtain the following by considering the adjoint picture in L_{1}. If τ_{∞} ⩾ −ω_{∞} for some ω_{∞} ∈ ℝ then we can associate a weak^{∗}- continuous semigroup T_{∞} on L_{∞} with the form τ, which satisfies ||T_{∞}(t)|| ⩽ e^{ω_{∞}t} for all t ⩾ 0. Observe that the condition on τ_{∞} imposes no restriction on b_{1}.

(f) Estimate (3.4) is analogous to the corresponding estimate in Theorem 1.32(b). In former results on second order elliptic operators with singular first order terms ([Lis96], [LiSe96]), an inequality similar to (3.4) was proved only for |u|^{\frac{p}{2}} in place of |u|^{\frac{p}{2}} sgn u. (Note that \overline{τ_{p}}(v) ⩾ \overline{τ_{p}}(|v|) for all v ∈ D(\overline{τ_{p}}) since \overline{τ_{p}} is a closed symmetric form fulfilling the first Beurling-Deny criterion.)

Corollary 3.4. Let the assumptions and notation be as in Theorem 3.2, and p ∈ I.

(a) Let U ⩾ 0 be τ_{a}-regular. Then U is T_{p}-regular, and τ + U ↔ (T_{p})_{U}.

(b) Let (U_{n})_{n∈ℕ_{0}} be a sequence of positive potentials such that U_{0} is τ_{a}-regular, U_{n} ⩽ U_{0}, τ +U_{n} is sectorial (n ∈ ℕ_{0}) and U_{n} → 0 a.e. as n → ∞. Then τ +U_{n} is closable, the analytic semigroup T_{U_{n},2} associated with \overline{τ + U_{n}} extrapolates to the C_{0}-semigroup (T_{p})_{U_{n}} on L_{p}, and U_{n} is T_{p}-regular. In particular, (T_{p})_{U_{n}} → T_{p} as n → ∞.

Proof. (a) Let W := W_{1} + W_{2} + V^{−}. Then τ + W is a closed sectorial form, by Proposition 3.1(a). By Lemma 1.24(b), U is (τ_{a} + W)-regular and hence (τ + W)-regular. By Proposition 1.21 we obtain the assertions of (a).

(b) Let n ∈ ℕ_{0}. By Proposition 3.1(a), τ + U_{n} + W is closed, so τ + U_{n} is closable by Corollary 1.16. By (a), U_{n} is T_{p}-regular, and τ + U_{n} ↔ (T_{p})_{U_{n}}, i.e., T_{U_{n},2} and (T_{p})_{U_{n}} are consistent. Finally, by [Voi88; Cor. 3.6], (T_{p})_{U_{n}} → T_{p} as n → ∞ since U_{0} is T_{p}-regular.

As a direct consequence of Theorem 3.2 we obtain a more explicit version of that theorem.

Corollary 3.5. Let V_{+}, V_{−} ⩾ 0 be τ_{a}-regular with V_{+} − V_{−} = V, and τ_{+} := Re τ_{a} + V_{+}. Assume that (a) and (bV) hold and that

(−1)^{j}⟨b_{j}u, ∇u⟩ ⩽ β_{j}τ_{+}(u) + B_{j}||u||_{2}^{2}, ⟨V_{−}u^{2}⟩ ⩽ γτ_{+}(u) + G||u||_{2}^{2}

(0 ⩽ u ∈ D(τ)∩Q(V_{+}), j = 1, 2) for some constants β_{1}, β_{2}, γ ⩾ 0, B_{1}, B_{2}, G ∈ ℝ. Let I_{0} := \{p ∈ [1, ∞); \frac{4}{pp′} − \frac{2}{p}β_{1} − \frac{2}{p′} β_{2} − γ ⩾ 0\}. Then, with the notation of Theorem 3.2, I ⊇ I_{0}, and ω_{p} ⩽ \frac{2}{p}B_{1} + \frac{2}{p′} B_{2} + G for all p ∈ I_{0}. Moreover, for all p ∈ ^{∘} I_{0} and u ∈ D(A_{p}) we have v_{p} := |u|^{\frac{p}{2}} sgn u ∈ D(τ_{+}) and

Re⟨A_{p}u, u|u|^{p−2}⟩ ⩾ (\frac{4}{pp′} − \frac{2}{p}β_{1} − \frac{2}{p′} β_{2} − γ)τ_{+}(v_{p}) − (\frac{2}{p}B_{1} + \frac{2}{p′} B_{2} + G)||u||_{p}^{p}.

If, in addition,

|Im⟨(b_{1} + b_{2})u, ∇u⟩| ⩽ c_{1}τ_{+}(u) + c_{2}||u||_{2}^{2} (u ∈ D(τ) ∩ Q(V_{+}))

for some c_{1} ⩾ 0, c_{2} ∈ ℝ then T_{p} extends to an analytic semigroup on L_{p} for all p ∈ ^{∘} I.

Proof. Since τ_{+}(|u|) ⩽ τ_{+}(u) for all u ∈ D(τ_{+}), and 1 ⩾ \frac{4}{pp′}, the assumptions imply that

τ_{p}(u) = \frac{4}{pp′} Re τ_{a}(u) + ⟨V_{+}|u|^{2}⟩ − (−\frac{2}{p}⟨b_{1}|u|, ∇|u|⟩) − \frac{2}{p′} ⟨b_{2}|u|, ∇|u|⟩ − ⟨V_{−}|u|^{2}⟩
⩾ (\frac{4}{pp′} − \frac{2}{p}β_{1} − \frac{2}{p′} β_{2} − γ)τ_{+}(u) − (\frac{2}{p}B_{1} + \frac{2}{p′} B_{2} + G)||u||_{2}^{2}

for all p ∈ [1, ∞), u ∈ D(τ) ∩ Q(V_{+}). Let W := W_{1} + W_{2} + |V |. Then τ_{p} is a bounded form on D(τ_{a} + W). Since V_{+} is (τ_{a} + W)-regular by Lemma 1.24(b), we deduce that τ_{p} ⩾ −(\frac{2}{p}B_{1} + \frac{2}{p′} B_{2} + G) for all p ∈ I_{0}. Thus, Theorem 3.2(a) implies the first two assertions. In order to obtain the remaining assertions, note that the above also implies that

τ_{p} ⩾ (\frac{4}{pp′} − \frac{2}{p}β_{1} − \frac{2}{p′} β_{2} − γ)τ_{+} − (\frac{2}{p}B_{1} + \frac{2}{p′} B_{2} + G)

for all p ∈ ^{∘} I_{0}.

Remarks 3.6. (a) The interval I_{0} defined in Corollary 3.5 can be non-empty only in the cases γ < 1, and γ = 1, β_{1} = β_{2} = 0 (then I_{0} = {2}).

(b) The assumptions of Corollary 3.5 are in particular fulfilled if W_{j} ⩽ β_{j}^{2}τ_{+} + 2β_{j}B_{j} for j = 1, 2 and V_{−} ⩽ γτ_{+} + G: then we have, by (3.2) and Euclid’s inequality,

|⟨b_{j}u, ∇u⟩| ⩽ \frac{β_{j}}{2} ⟨a_{s}∇u, ∇u⟩ + \frac{1}{2β_{j}} ⟨W_{j}|u|^{2}⟩

⩽ \frac{β_{j}}{2} Re τ_{a}(u) + \frac{1}{2β_{j}} (β_{j}^{2}τ_{+}(u) + 2β_{j}B_{j}||u||_{2}^{2}) ⩽ β_{j}τ_{+}(u) + B_{j}||u||_{2}^{2}

and thus also

|Im⟨(b_{1} + b_{2})u, ∇u⟩| ⩽ (β_{1} + β_{2})τ_{+}(u) + (B_{1} + B_{2})||u||_{2}^{2}

for all u ∈ D(τ) ∩ Q(V_{+}). In this way, we reobtain [Lis96; Thms. 1-5] as special cases of Corollary 3.5.

We point out that, besides some additional restrictions, b_{2} was assumed to be 0 in [Lis96]. In this case, the semigroup associated with τ +V^{−} is L_{∞}-contractive (see Remark 3.3(e)). This leads to considerable simplifications in the proofs. However, if b_{2} ≠ 0 then it is not clear whether there exists a τ_{a}-regular potential U such that the semigroup associated with τ + U is L_{∞}-bounded.

One of the major disadvantages of the assumption W_{j} ⩽ β_{j}^{2}τ_{+} + 2β_{j}B_{j} (in comparison with the corresponding assumption in Corollary 3.5) is that it does not respect the sign of the drift: if the assumption is fulfilled for b_{j} then also for −b_{j}.

(c) The example of the Ornstein-Uhlenbeck operator L = −Δ + Bx · ∇ shows that the conditions posed on b_{1}, b_{2} in Corollary 3.5 are much less restrictive than the conditions studied in (b). Let Ω = ℝ^{N}, a = id, D(τ_{a}) = W_{2}^{1}(ℝ^{N}). Let V = 0, b_{2} = 0, and define b_{1} by b_{1}(x) = Bx for some B ∈ ℝ^{N} ⊗ ℝ^{N}. Then W_{1} = |b_{1}|^{2} is τ_{a}-regular. Moreover,

−⟨b_{1}∇u, u⟩ = −\frac{1}{2}⟨b_{1}, ∇u^{2}⟩ = \frac{1}{2} tr B||u||_{2}^{2} (0 ⩽ u ∈ C_{c}^{1}(ℝ^{N})),

so we obtain that the assumptions of Corollary 3.5 hold with V_{+} = V_{−} = 0, β_{1} = β_{2} = B_{2} = γ = G = 0, B_{1} = \frac{1}{2} tr B. Hence, τ is associated with a consistent family of positive quasi-contractive C_{0}-semigroups T_{p} on L_{p}, p ⩾ 1, with ||T_{p}(t)|| ⩽ e^{\frac{t}{p} tr B} for all t ⩾ 0, p ⩾ 1.

Now we show that the conditions posed in (b) are not fulfilled with γ < 1 (cf. (a)) unless B = 0. Assume that there exist β_{1} ⩾ 0, γ ∈ [0, 1), B_{1}, G ∈ ℝ, V_{+} = V_{−} measurable (so that V_{+} − V_{−} = 0) such that W_{1} ⩽ β_{1}^{2}(τ_{a} + V_{+}) + 2β_{1}B_{1} and V_{−} ⩽ γ(τ_{a} + V_{+}) + G. Then (1 − γ)V_{+} ⩽ γτ_{a} + G and hence W_{1} ⩽ c(τ_{a} + 1) for some c > 0. If B ≠ 0 then W_{1} = |b_{1}|^{2} increases at infinity, in some direction x. More precisely, there exists x ∈ ℝ^{N} such that W_{1} ⩾ λ^{2} on B(λx, 1), for all λ ⩾ 1. It is easy so see that this contradicts W_{1} ⩽ c(τ_{a} + 1).

For further examples, in particular where τ is associated with a semigroup on L_{p} only for p from some subinterval of [1, ∞), see Section 3.4.

In the remainder of the section we make the following assumption (cf. [Sem97; Def. 3.1]).

(BC) For all 0 ⩽ φ ∈ W_{∞}^{1} (i.e., φ bounded and Lipschitz continuous) that satisfy

a_{s}∇φ · ∇φ ⩽ c(Re τ_{a} + W_{1} + W_{2} + V^{+} + 1) for some c > 0,

u ∈ D(τ) implies φu ∈ D(τ).

The above assumption is a restriction on the type of boundary conditions. Below we show that it holds in the case of Neumann and of Dirichlet boundary conditions. However, (BC) does not hold for periodic type boundary conditions. In the case a ∈ L_{∞}, (BC) simply reads φu ∈ D(τ) for all 0 ⩽ φ ∈ W_{∞}^{1}, u ∈ D(τ). Thus, if a ∈ L_{∞} then we have the following. Assume that (BC) holds, τ̃_{a} ⊆ τ_{a} is a Dirichlet form, and that D(τ̃_{a}) is an ideal of D(τ_{a}) (u ∈ D(τ̃_{a}), v ∈ D(τ_{a}) and |v| ⩽ |u| imply v ∈ D(τ̃_{a})). Then (BC) holds with τ̃_{a} in place of τ_{a}.

Proposition 3.7. ([LiVo00; Prop. 9]) Let (a) and (bV) hold. If τ_{a} = τ_{N} or τ_{a} = τ_{D} (see Propositions 1.28 and 1.30) then assumption (BC) is fulfilled.

Proof. Let W := W_{1} + W_{2} + V^{+} + 1. Recall that D(τ) = D(τ_{a} + W).

Let first τ_{a} = τ_{N}. Let 0 ⩽ φ ∈ W_{∞}^{1} with a_{s}∇φ · ∇φ ⩽ c(Re τ_{N} + W) for some c > 0, and 0 ⩽ u ∈ D(τ_{N} + W). Then, by Euclid’s inequality,

\begin{aligned}a_{s}∇(φu) · ∇(φu) & = φ^{2}a_{s}∇u · ∇u + u^{2}a_{s}∇φ · ∇φ + 2φua_{s}∇φ · ∇u \\ & ⩽ 2||φ||_{∞}^{2}a_{s}∇u · ∇u + 2u^{2}a_{s}∇φ · ∇φ ∈ L_{1}. \end{aligned}  (3.6)

Hence φu ∈ D(τ_{N}), and it is clear that φu ∈ Q(W). This proves the assertion for τ_{a} = τ_{N} since τ_{N} + W fulfils the first Beurling-Deny criterion.

Now let τ_{a} = τ_{D}. Let 0 ⩽ φ ∈ W_{∞}^{1} with a_{s}∇φ · ∇φ ⩽ c(Re τ_{D} + W) for some c > 0 (this does not imply a_{s}∇φ · ∇φ ⩽ c(Re τ_{N} + W)!). Obviously, D(τ_{D}) ∩ L_{∞,c} is a dense ideal of D(τ_{D}). Thus, by Lemma 1.24(a), D(τ_{D} + W) ∩ L_{∞,c} =: D_{∞,c} is dense in D(τ_{D} + W). Let 0 ⩽ u ∈ D_{∞,c}. Below we show that φu ∈ D(τ_{D}). Then we obtain, by (3.6) and the assumption on φ,

(τ_{D} + W)(φu) ⩽ 2||φ||_{∞}^{2}τ_{D}(u) + 2⟨a_{s}∇φ · ∇φ, u^{2}⟩ + ||φ||_{∞}^{2}⟨W u^{2}⟩
⩽ 2(||φ||_{∞}^{2} + c)(τ_{D} + W)(u).

Since D_{∞,c} is a lattice and τ_{D} + W is a Dirichlet form, we conclude that φ acts as a bounded multiplication operator from D_{∞,c} ⊆ D(τ_{D} + W) to D(τ_{D} + W). This yields the assertion since D_{∞,c} is dense in D(τ_{D} + W).

So, let 0 ⩽ u ∈ D_{∞,c}. We have to show φu ∈ D(τ_{D}). Let (v_{n}) ⊆ C_{c}^{∞} such that v_{n} → u in D(τ_{D}) as n → ∞. Let u_{n} := |v_{n}| ∧ ||u||_{∞} (∈ W_{∞,c}^{1}). Then sup_{n∈ℕ} τ_{D}(u_{n}) ⩽ sup_{n∈ℕ} Re τ_{D}(v_{n}) < ∞ since τ_{D} is a Dirichlet form. Let ψ ∈ C_{c}^{∞} such that ψu = u. Let w_{n} := φψu_{n}. Then φψ, w_{n} ∈ W_{∞,c}^{1} ⊆ D(τ_{D}) by Proposition 1.30. By (3.6) we obtain that

τ_{D}(w_{n}) ⩽ 2||φψ||_{∞}^{2}τ_{D}(u_{n}) + 2||u_{n}||_{∞}^{2}τ_{D}(φψ),

which implies that sup_{n∈ℕ} τ_{D}(w_{n}) < ∞ (recall that ||u_{n}||_{∞} ⩽ ||u||_{∞}). Further, w_{n} → φψu = φu in L_{2} as n → ∞. Hence φu ∈ D(τ_{D}), by the lower semicontinuity of τ_{D}.

The following is our main result on extrapolation for second order elliptic differential operators.

Theorem 3.8. Let (a), (bV) and (BC) hold, and N ⩾ 3. Let I and T_{p} (p ∈ I) be defined as in Theorem 3.2. Assume that

(i) a ∈ L_{∞}, and there exist p ∈ ^{∘} I, ε_{p} > 0, c_{p} ∈ ℝ such that the form τ_{p} admits the Sobolev imbedding

τ_{p}(u) ⩾ ε_{p}||u||2_{\frac{2N}{N−2}} − c_{p}||u||_{2}^{2} (u ∈ D(τ));

(ii) there exist p ∈ ^{∘} I, C_{p} ⩾ 0 such that T_{p} is analytic, and

|⟨(b_{1} + b_{2})u^{2}⟩| ⩽ C_{p}||u||_{τ_{p}}||u||_{2} (0 ⩽ u ∈ D(τ)).

Let p_{+} := sup I, p_{−} := inf I, p_{max} := \frac{N}{N−2}p_{+}, p_{min} := (\frac{N}{N−2}p′_{−})′. In case 1 ∈ I let I_{max} := [1, p_{max}), otherwise I_{max} := (p_{min}, p_{max}). Then τ is associated with an analytic semigroup T_{p}(t) = e^{−tA_{p}} on L_{p}, for all p ∈ I_{max}. The angle of analyticity of T_{p} and the spectrum σ(A_{p}) are independent of p ∈ I_{max}.

Remarks 3.9. (a) Note that, by Proposition 3.1(c) and Stein interpolation, both assumptions (i) and (ii) of the above theorem are fulfilled for some p ∈ ^{∘} I if and only if they are fulfilled for all p ∈ ^{∘} I.

(b) Concerning assumption (i), the reader should think of a uniformly elliptic matrix function a. Then, by Proposition 3.1(b), (i) is fulfilled if either of the following three conditions holds: D(τ_{a}) ⊆ W_{2,0}^{1}, Ω has the cone property, or Ω has the extension property (see [Ada75]).

(c) Concerning assumption (ii), note that it is much less restrictive to pose a condition on |⟨(b_{1} + b_{2})u^{2}⟩| than on ⟨|b_{1} + b_{2}|u^{2}⟩.

Assumption (ii) is in particular fulfilled if a is uniformly elliptic and

|b_{1} + b_{2}|^{2} ⩽ K_{p}τ_{p} + ω̃_{p}

for some K_{p} > 0, ω̃_{p} ∈ ℝ. To see this, first observe that the latter condition is equivalent to ||(b_{1} + b_{2})u||_{2} ⩽ k_{p}||u||_{τ_{p}} for some k_{p} > 0 and all u ∈ D(τ_{p}).

In order to show that T_{p} is analytic, we check condition (3.5) of Theorem 3.2. For all u ∈ D(τ) we have

|⟨(b_{1} + b_{2})u, ∇u⟩| ⩽ ||(b_{1} + b_{2})u||_{2}||∇u||_{2} ⩽ k_{p}||u||_{τ_{p}}||∇u||_{2}.

By Proposition 3.1(b) and the uniform ellipticity of a, there exists c > 0 such that ||∇u||_{2} ⩽ c||u||_{τ_{p}} for all u ∈ D(τ_{p}). Thus,

|⟨(b_{1} + b_{2})u, ∇u⟩| ⩽ k_{p}c||u||_{τ_{p}}^{2} = k_{p}c(τ_{p}(u) + (ω_{p} + 1)||u||_{2}^{2}) (u ∈ D(τ)).

By Theorem 3.2(c) we infer that T_{p} is analytic.

Moreover, for all 0 ⩽ u ∈ D(τ) we have

|⟨(b_{1} + b_{2})u^{2}⟩| ⩽ ||(b_{1} + b_{2})u||_{2}||u||_{2} ⩽ k_{p}||u||_{τ_{p}}||u||_{2},

i.e., assumption (ii) of Theorem 3.8 is fulfilled.

(d) In the cases N = 1, 2, it should be possible to prove an analogue of Theorem 3.8 with I_{max} = (1, ∞) or even I_{max} = [1, ∞). Of course assumption (i) has to be reformulated for N = 1, 2.

We conclude the section by a result on L_{p}-spectral independence which is a generalisation of [LiVo00; Thm. 2]. Recall the notion of L_{1}-regularity from Section 2.2, (2.11).

Theorem 3.10. Let (a), (bV) and (BC) hold. Let I and T_{p}(t) = e^{−tA_{p}} (p ∈ I) be defined as in Theorem 3.2. Assume that there exist p ∈ ^{∘} I, r > 1, ε_{p} > 0, c_{p} ∈ ℝ such that the form τ_{p} admits the Sobolev imbedding

τ_{p}(u) ⩾ ε_{p}||u||_{2r}^{2} − c_{p}||u||_{2}^{2} (u ∈ D(τ)).

3.2. QUASI-CONTRACTIVE C_{0}-SEMIGROUPS 67

(a) Assume that there exists an L_{1}-regular function ψ = (ψ_{1}, …, ψ_{N}): ℝ^{N} → ℝ^{N} fulfilling the following two inequalities for some p ∈ ^{∘} I, c_{0} ⩾ 0, c_{1} ∈ ℝ, for all j = 1, …, N:

a∇ψ_{j} · ∇ψ_{j} ⩽ c_{0}τ_{p} + c_{1},
|⟨(b_{1} + b_{2}) · ∇ψ_{j}, u^{2}⟩| ⩽ c_{0}τ_{p}(u) + c_{1}||u||_{2}^{2} (0 ⩽ u ∈ D(τ)).

Then the spectrum σ(A_{p}) is independent of p ∈ I \{1}, and the operators A_{p} have consistent resolvents.

(b) If, more restrictively than in (a), there exist c > 0, ε ∈ (0, 2], p ∈ ^{∘} I such that

⟨a∇ψ_{j} · ∇ψ_{j}, u^{2}⟩ + |⟨(b_{1} + b_{2}) · ∇ψ_{j}, u^{2}⟩| ⩽ c||u||_{τ_{p}}^{2−ε}||u||_{2}^{ε}

for all 0 ⩽ u ∈ D(τ), j = 1, …, N then σ(A_{p}) is independent of p ∈ I, and the operators A_{p} have consistent resolvents.

Remarks 3.11. (a) Note that the function ψ(x) = x is L_{1}-regular, and ∇ψ_{j} = e_{j} for all j = 1, …, N, where e_{j} are the standard unit vectors of ℝ^{N}. Thus, the assumptions of Theorem 3.10 are much weaker than the assumptions of Theorem 3.8. In particular, we do not need to assume the matrix function a to be bounded—if ψ(x) = x then the first condition in Theorem 3.10(a) is just form boundedness of a_{jj} with respect to τ_{p}, for all j = 1, …, N. The attentive reader will note that the latter is a self-referential condition on a. It allows a to have strong local singularities.

(b) The following example reveals the relevance of the notion of L_{1}-regularity: Let K_{n} ≠ ∅ (n ∈ ℕ) be compact subsets of ℝ^{N} such that

sup_{n∈ℕ}diam(K_{n}) < ∞, inf_{n≠m} dist(K_{n}, K_{m}) > 0.

For n ∈ ℕ fix some x_{n} ∈ K_{n}. Define the function ψ_{0} on ⋃_{n∈ℕ} K_{n} by ψ_{0}(x) := x_{n} for all n ∈ ℕ, x ∈ K_{n}. Then ψ_{0} is Lipschitz continuous.

By Kirszbraun’s theorem (see, e.g., [Fed96; 2.10.43.]), ψ_{0} has a Lipschitz continuous extension ψ to ℝ^{N}. Without restriction assume that for all j ∈ ℤ^{N} there exists n ∈ ℕ such that dist(j, K_{n}) ⩽ 1 (otherwise add {j} to the collection of K_{n}). Then it is easy to see that sup_{j∈ℤ^{N}} |ψ(j) − j| < ∞. From this we deduce that ψ is L_{1}-regular. By the construction, ∇ψ_{j}↾_{K_{n}} = 0 for all n ∈ ℕ, j = 1, …, N. Thus, by the assumptions on a∇ψ_{j} · ∇ψ_{j} and (b_{1} + b_{2}) · ∇ψ_{j} in Theorem 3.10 we pose no restrictions on the coefficients a, b_{1}, b_{2} on ⋃_{n∈ℕ} K_{n}.

3.2 Quasi-contractive C_{0}-semigroups

In this section we prove Theorem 3.2. We separate the core of the proof into a lemma. Fix p ∈ (1, ∞). For u ∈ L_{1,loc}, n ∈ ℕ let u_{n,p} := (|u|^{\frac{p}{2} −1}) ∧ n, v_{n,p} := uu_{n,p}, w_{n,p} := uu_{n,p}^{2}, and v_{p}(u) := u|u|^{\frac{p}{2} −1}, w_{p}(u) := u|u|^{p−2} as in Lemma 1.34.

Lemma 3.12. Let τ be a densely defined sesquilinear form in L_{2} fulfilling the first Beurling-Deny criterion. Let 𝔥 be a closed symmetric form in L_{2}, 𝔥 ⩾ −ω for some ω ∈ ℝ. Assume that there exists a sequence (U_{n})_{n∈ℕ_{0}} of positive potentials such that D(U_{0}) ⊇ D(τ), τ + U_{0} is sectorial and closed, U_{n} ↓ 0 as n → ∞, and

w_{n,p} ∈ D(τ), v_{n,p} ∈ D(𝔥), Re τ(u, w_{n,p}) ⩾ 𝔥(v_{n,p}) − ⟨U_{n}|v_{n,p}|^{2}⟩  (3.7)

for all u ∈ D(τ), n ∈ ℕ.

(a) Then τ is associated with a positive C_{0}-semigroup T_{p}(t) = e^{−tA_{p}} on L_{p} with ||T_{p}(t)|| ⩽ e^{ωt} (t ⩾ 0), and for all u ∈ D(A_{p}) we have

Re⟨A_{p}u, w_{p}(u)⟩ ⩾ 𝔥(v_{p}(u)).  (3.8)

(b) If, in addition,

|Im τ(u, w_{n,p})| ⩽ M(Re τ + U_{n} + ω̃)(u, w_{n,p}) (u ∈ D(τ), n ∈ ℕ)

for some M ⩾ 0, ω̃ ∈ ℝ, then A_{p} is m-sectorial of angle arctan M. In particular, T_{p} is an analytic semigroup.

Proof. (a) Without restriction assume ω = 0. Let A_{0} be the m-sectorial operator in L_{2} associated with τ + U_{0}. In a first step we show that e^{−tA_{0}} extrapolates to a contractive C_{0}-semigroup T_{0,p}(t) = e^{−tA_{0,p}} on L_{p}. Then we make use of Lemma 1.34 to prove the assertions of (a).

(i) By the exponential formula, it suffices to show that, given f ∈ L_{2} ∩ L_{p} and 0 < λ ∈ ρ(−A_{0}), one has ||(λ + A_{0})^{−1}f||_{p} ⩽ \frac{1}{λ}||f||_{p}. Let u := (λ + A_{0})^{−1}f. Then u ∈ D(τ + U_{0}) = D(τ). This implies that v_{n,p} ∈ Q(U_{0}). By assumption (3.7) and the equality u\overline{w}_{n,p} = |v_{n,p}|^{2} we have, for all n ∈ ℕ,

\begin{aligned}λ||v_{n,p}||_{2}^{2} + (𝔥 + U_{0} − U_{n})(v_{n,p}) & ⩽ λ⟨u, w_{n,p}⟩ + Re(τ + U_{0})(u, w_{n,p}) \\ & = Re⟨(λ + A_{0})u, w_{n,p}⟩ ⩽ ||f||_{p}||w_{n,p}||_{p′}. \end{aligned} (3.9)

Observe that |w_{n,p}|^{p′} = |u|^{p′}u_{n,p}^{2p′} ⩽ |u|^{2}u_{n,p}^{2} = |v_{n,p}|^{2}. Hence ||w_{n,p}||_{p′} ⩽ ||v_{n,p}||2^{\frac{2}{p′}},

and from (3.9) we obtain, noting 𝔥 + U_{0} − U_{n} ⩾ 𝔥 ⩾ 0,

||v_{n,p}||2^{\frac{2}{p}} ⩽ \frac{1}{λ}||f||_{p} (n ∈ ℕ).

Since |v_{n,p}| ↑ |v_{p}(u)| we conclude by monotone convergence that v_{p}(u) ∈ L_{2}, and

||(λ + A_{0})^{−1}f||_{p} = ||u||_{p} = ||v_{p}(u)||2^{\frac{2}{p}} ⩽ \frac{1}{λ}||f||_{p}.

(ii) With the quantities introduced in (i) we proceed as follows. By dominated convergence, v_{n,p} → v_{p}(u) in L_{2} and w_{n,p} → w_{p}(u) in L_{p′} as n → ∞. Further, A_{0}u = f − λu ∈ L_{p}. From estimate (3.9) we obtain

lim inf_{n→∞}(𝔥(v_{n,p}) + ⟨(U_{0} − U_{n})|v_{n,p}|^{2}⟩) ⩽ lim_{n→∞} Re⟨A_{0}u, w_{n,p}⟩ = Re⟨A_{0}u, w_{p}(u)⟩.

By monotone convergence, (U_{0} − U_{n})|v_{n,p}|^{2} ↑ U_{0}|v_{p}(u)|^{2} in L_{1}. Hence, the left hand side of the previous inequality equals lim inf_{n→∞} 𝔥(v_{n,p}) + ⟨U_{0}|v_{p}(u)|^{2}⟩. The lower semicontinuity of 𝔥 implies that

v_{p}(u) ∈ D(𝔥), (𝔥 + U_{0})(v_{p}(u)) ⩽ Re⟨A_{0}u, w_{p}(u)⟩.  (3.10)

So far we have proved inequality (3.10) for all u from the core D := (λ+A_{0})^{−1}(L_{2}∩ L_{p}) of A_{0,p}, where λ > 0 is some element of ρ(−A_{0}).

Let now u ∈ D(A_{0,p}). Choose (u^{(m)}) ⊆ D such that u^{(m)} → u in D(A_{0,p}). Then v_{p}(u^{(m)}) → v_{p}(u) in L_{2} and w_{p}(u^{(m)}) → w_{p}(u) in L_{p′}. By (3.10) we obtain

lim inf_{m→∞}(𝔥 + U_{0})(v_{p}(u^{(m)})) ⩽ lim_{m→∞} Re⟨A_{0,p}u^{(m)}, w_{p}(u^{(m)})⟩ = Re⟨A_{0,p}u, w_{p}(u)⟩,

so the lower semicontinuity of 𝔥+U_{0} implies that (3.10) holds for all u ∈ D(A_{0,p}).

Now we are in a position to apply Lemma 1.34, with 𝔥 + U_{0} in place of 𝔥, A = A_{0,p}, and V = −U_{0}. By (3.10) we infer that −U_{0} is T_{0,p}-admissible, that T_{p} := (T_{0,p})_{−U_{0}} is a contractive C_{0}-semigroup, and that (3.8) holds, with −A_{p} the generator of T_{p}.

(b) Recall that τ + U_{0} ↔ e^{−tA_{0,p}}. For m ∈ ℕ let A_{m} := A_{0,p} − U_{0} ∧ m. Then τ + (U_{0} − m)^{+} ↔ e^{−tA_{m}}. Let u ∈ D(A_{0}) ∩ D(A_{0,p}). Then, since u\overline{w}_{n,p} = |v_{n,p}|^{2} is real and A_{0,p}u = A_{0}u,

Im⟨A_{m}u, w_{n,p}⟩ = Im⟨(A_{0} − U_{0} ∧ m)u, w_{n,p}⟩ = Im τ(u, w_{n,p}) (m, n ∈ ℕ). (3.11)

By (3.10) we know that U_{n}|u\overline{w}_{n,p}| ⩽ U_{0}|v_{p}(u)|^{2} ∈ L_{1}. Thus, ⟨U_{n}u, w_{n,p}⟩ → 0 by dominated convergence. By (3.11) and the assumption of (b) we conclude that

|Im⟨A_{m}u, w_{p}(u)⟩| = lim_{n→∞} |Im τ(u, w_{n,p})|

⩽ lim_{n→∞} M(Re τ + (U_{0} − m)^{+} + U_{n} + ω̃)(u, w_{n,p}) = M Re⟨(A_{m} + ω̃)u, w_{p}(u)⟩.

This estimate carries over to all u ∈ D(A_{m}) since D(A_{0}) ∩ D(A_{0,p}) is a core for A_{m}.

Let now u ∈ D(A_{p}). We have A_{m} → A_{p} in the strong resolvent sense. Thus, u^{(m)} := (1 + A_{m})^{−1}(1 + A_{p})u → u in L_{p} and w_{p}(u^{(m)}) → w_{p}(u) in L_{p′} as m → ∞. Since

u^{(m)} + A_{m}u^{(m)} = u + A_{p}u,

we also have A_{m}u^{(m)} → A_{p}u in L_{p} as m → ∞. Hence,

|Im⟨A_{p}u, w_{p}(u)⟩| = lim_{m→∞} |Im⟨A_{m}u^{(m)}, w_{p}(u^{(m)})⟩|

⩽ lim_{m→∞} M Re⟨(A_{m} + ω̃)u^{(m)}, w_{p}(u^{(m)})⟩ = M Re⟨(A_{p} + ω̃)u, w_{p}(u)⟩,

which shows the m-sectoriality of A_{p}, with angle arctan M.

From the proof we easily see: In order to show τ ↔ T_{p} on L_{p} with ||T_{p}(t)|| ⩽ e^{ωt} it suffices to require τ(u, w_{n,p}) + ⟨U_{n}v_{n,p}^{2}⟩ ⩾ −ω for all 0 ⩽ u ∈ D(τ), n ∈ ℕ.

In order to use Lemma 3.12 in the proof of Theorem 3.2, we need to know that v_{n,p} and w_{n,p} are multiples of normal contractions of u. This is a consequence of the following lemma.

Lemma 3.13. Let φ: [0, ∞) → ℝ such that φ(0) = 0 and |φ(s) − φ(t)| ⩽ |s − t| for all s, t ⩾ 0. Then φ̂(z) := φ(|z|) sgn z defines a normal contraction φ̂ on ℂ.

Proof. Let s, t ⩾ 0, α, β ∈ [0, 2π). Then we have

|φ̂(se^{iα}) − φ̂(te^{iβ})|^{2} = |φ(s)e^{iα} − φ(t)e^{iβ}|^{2} = φ(s)^{2} + φ(t)^{2} − 2φ(s)φ(t) Re e^{iα−iβ}
= (φ(s) − φ(t))^{2} + 2φ(s)φ(t)(1 − Re e^{iα−iβ})
⩽ (s − t)^{2} + 2st(1 − Re e^{iα−iβ}) = |se^{iα} − te^{iβ}|^{2}.

In the application to the functions v_{n,p}, w_{n,p}, the function φ is of the type

φ_{α,r}(x) = x(x^{α} ∧ r) = \{ \begin{matrix}x^{α+1} & if x^{α} < r, \\ rx & if x^{α} ⩾ r, \end{matrix}

with α ∈ ℝ, r > 0. Here, x^{0} := 1 for all x ⩾ 0. In the next lemma we compute the gradient of φ̂_{α,r} ∘ u for u ∈ W_{1,loc}^{1}.

Lemma 3.14. Let α ∈ ℝ, r > 0, u ∈ W_{1,loc}^{1}, u_{α,r} := |u|^{α} ∧ r. Then uu_{α,r} = φ̂_{α,r} ∘ u is a multiple of a normal contraction of u, uu_{α,r} ∈ W_{1,loc}^{1}, and

∇(uu_{α,r}) = u_{α,r}(∇u + α sgn u · χ_{[|u|^{α}<r]}∇|u|).

Proof. The first assertion follows from Lemma 3.13 and the Lipschitz continuity of the function φ_{α,r} defined above. In the case α ∉ (0, 1), the function [0, ∞) ∋ x ↦ x^{α} ∧ r is Lipschitz continuous, hence u_{α,r} ∈ W_{1,loc}^{1}, so the remaining assertions follow by an application of the product rule and the chain rule (for the latter see, e.g., [BoMu82; appendix]—the proof given there for u ∈ W_{2,0}^{1} works also for u ∈ W_{1,loc}^{1}).

In the case α ∈ (0, 1), we approximate |u| by u_{δ} := |u| + δ (δ > 0). Note that (u_{δ}^{α} ∧ r)∇u ∈ L_{1,loc} and

u∇(u_{δ}^{α} ∧ r) = αuu_{δ}^{α−1}χ_{[u_{δ}^{α}<r]}∇u_{δ} = (u_{δ}^{α} ∧ r) · α\frac{u}{u_{δ}} χ_{[u_{δ}^{α}<r]}∇|u| ∈ L_{1,loc}.

By the product rule we obtain that

∇(u(u_{δ}^{α} ∧ r)) = (u_{δ}^{α} ∧ r)(∇u + α\frac{u}{u_{δ}} χ_{[u_{δ}^{α}<r]}∇|u|).

Finally, u(u_{δ}^{α} ∧ r) → uu_{α,r} and

∇(u(u_{δ}^{α} ∧ r)) → u_{α,r}(∇u + α sgn u · χ_{[|u|^{α}<r]}∇|u|)

in L_{1,loc} as δ → 0. This implies the assertion.

Proof of Theorem 3.2. Let p ∈ I, i.e., τ_{p} ⩾ −ω_{p}. Let U_{0} := W_{1} + W_{2} + V^{−}. By Proposition 3.1(a), τ + U_{0} is a closed sectorial form.

First we study the case p > 1. Let u ∈ D(τ). Then v_{n,p}, w_{n,p} ∈ D(τ) as multiples of normal contractions of u. At the end of the proof we will show that

Re τ(u, w_{n,p}) ⩾ τ_{p}(v_{n,p}) − \frac{1}{2}⟨χ_{n}(W_{1} + W_{2})|v_{n,p}|^{2}⟩,  (3.12)

where χ_{n} is the indicator of the set [|u|^{\frac{p−2}{2}} ⩾ n]. Applying Lemma 3.12(a) with 𝔥 = \overline{τ_{p}} and U_{n} = \frac{1}{2}χ_{n}(W_{1} + W_{2}) (n ∈ ℕ), we obtain all the assertions of Theorem 3.2(a) and (b).

Let now assumption (3.5) hold for some p ∈ ^{∘} I. Then it holds for all p ∈ ^{∘} I, by Proposition 3.1(b). To prove the analyticity of T_{p} we need the inequality

|Im τ(u, w_{n,p})| ⩽ |Im τ_{a}(v_{n,p})| + |\frac{1}{p} − \frac{1}{p′} | Re τ_{a}(v_{n,p}) + |Im⟨(b_{1} + b_{2})v_{n,p}, ∇v_{n,p}⟩|, (3.13)

which is also shown at the end of the proof. The first term on the right hand side of (3.13) can be estimated by α Re τ_{a}(v_{n,p}), due to assumption (a). Thus, by (3.5) we obtain that

|Im τ(u, w_{n,p})| ⩽ (α + |\frac{1}{p} − \frac{1}{p′} |) Re τ_{a}(v_{n,p}) + c_{1}τ_{p}(v_{n,p}) + c_{2}||v_{n,p}||^{2}.

By Proposition 3.1(b) and estimate (3.12) we have

Re τ_{a}(v_{n,p}) ⩽ C(τ_{p} + ω̃_{1})(v_{n,p}) ⩽ C(Re τ + U_{n} + ω̃_{1})(u, w_{n,p})

for some ω̃_{1} ∈ ℝ, C > 0 depending on p. We conclude that

|Im τ(u, w_{n,p})| ⩽ [C(α + |\frac{1}{p} − \frac{1}{p′} |) + c_{1}](Re τ + U_{n} + ω̃_{2})(u, w_{n,p})

for some ω̃_{2} ∈ ℝ, so Lemma 3.12(b) implies that A_{p} is an m-sectorial operator.

The proof for the case p = 1 is based on the assertions of the theorem in the case p > 1, applied to the form τ̃ := τ + U_{0}, with U_{0} as above. Recall that τ̃ is a closed sectorial form in L_{2}. Let T_{0} be the associated analytic semigroup on L_{2}. Let 1 < p < ∞ and τ̃_{p} := τ_{p} + U_{0}. For all 0 ⩽ u ∈ D(τ̃) = D(τ) we have

τ̃_{p}(u) = \frac{4}{pp′} τ_{a}(u) − \frac{2}{p′} ⟨b_{2}u, ∇u⟩ + \frac{1}{p}(2⟨∇u, b_{1}u⟩ + ⟨V u^{2}⟩) + ⟨(\frac{1}{p′} V + U_{0})u^{2}⟩.

We apply Euclid’s inequality to the second term, and the estimate

τ_{1}(u) = 2⟨∇u, b_{1}u⟩ + ⟨V u^{2}⟩ ⩾ −ω_{1}||u||_{2}^{2}

to the third term on the right hand side, to obtain

τ̃_{p}(u) ⩾ \frac{4}{pp′} τ_{a}(u) − \frac{2}{p′} (\frac{1}{2}τ_{a}(u) + \frac{1}{2}⟨W_{2}u^{2}⟩) − \frac{ω_{1}}{p} ||u||_{2}^{2} + ⟨(U_{0} − \frac{1}{p′} V^{−})u^{2}⟩
= \frac{1}{p′} (\frac{4}{p} − 1)τ_{a}(u) − \frac{ω_{1}}{p} ||u||_{2}^{2} + ⟨(U_{0} − \frac{1}{p′} (V^{−} + W_{2}))u^{2}⟩.

For 1 < p ⩽ 4, Theorem 3.2(a) and (b) applied to τ̃ imply: T_{0} extrapolates to a C_{0}-semigroup T_{0,p} on L_{p}, and for the generator −A_{0,p} of T_{0,p} we have

⟨A_{0,p}u, u^{p−1}⟩ ⩾ ⟨(U_{0} − \frac{1}{p′} (V^{−} + W_{2}))u^{p}⟩ − \frac{ω_{1}}{p} ||u||_{p}^{p} (0 ⩽ u ∈ D(A_{0,p})). (3.14)

In particular, ||T_{0,p}(t)||_{p→p} ⩽ e^{\frac{ω_{1}}{p} t} for all t ⩾ 0, 1 < p ⩽ 4 (recall U_{0} = W_{1} + W_{2} + V^{−}). Since T_{0} is a positive C_{0}-semigroup, [Voi92] implies that T_{0} extrapolates to a C_{0}-semigroup T_{0,1} on L_{1}.

Let now U_{n,m} := (U_{0} − \frac{1}{m}(V^{−} + W_{2})) ∧ n for n, m ∈ ℕ. It follows from (3.14) that

||(T_{0,p})_{−U_{n,m}}(t)||_{p→p} ⩽ e^{\frac{ω_{1}}{p} t} (t ⩾ 0)

for all n ∈ ℕ, m ⩾ 2 and 1 < p ⩽ \frac{m}{m−1} (i.e., \frac{1}{p′} ⩽ \frac{1}{m}). Since (T_{0,p})_{−U_{n,m}} and (T_{0,1})_{−U_{n,m}} are consistent by Lemma 1.10(b), we obtain ||(T_{0,1})_{−U_{n,m}}(t)||_{1→1} ⩽ e^{ω_{1}t} for all t ⩾ 0, n ∈ ℕ, m ⩾ 2. Since U_{n,m} ↑ U_{0} ∧ n as m → ∞, we have (T_{0,1})_{−U_{n,m}} → (T_{0,1})_{−U_{0}∧n} for all n ∈ ℕ, by [Voi86; Prop. A.2]. Hence

sup_{n∈ℕ}||(T_{0,1})_{−U_{0}∧n}(t)||_{1→1} ⩽ e^{ω_{1}t} (t ⩾ 0).

Finally, [Voi88; Prop. 2.2] implies that −U_{0} is T_{0,1}-admissible, and we obtain τ ↔ (T_{0,1})_{−U_{0}} =: T_{1}, with ||T_{1}(t)||_{1→1} ⩽ e^{ω_{1}t} for all t ⩾ 0.

To complete the proof it remains to show inequalities (3.12) and (3.13). Let χ_{n}^{c} := 1 − χ_{n}, i.e., the indicator of the set [|u|^{\frac{p−2}{2}} < n]. We write u_{n} = u_{n,p}, v_{n} = v_{n,p} (= u(|u|^{\frac{p−2}{2}} ∧n)) and w_{n} = w_{n,p} (= u(|u|^{p−2}∧n^{2})) for short. Lemma 3.14 implies that

∇v_{n} = u_{n}(∇u + \frac{p−2}{2}χ_{n}^{c} sgn u∇|u|) = sgn u(u_{n} sgn \overline{u}∇u + \frac{p−2}{2}χ_{n}^{c}u_{n}∇|u|).

Let φ_{n} := u_{n} Re(sgn \overline{u}∇u) = u_{n}∇|u| and ψ_{n} := u_{n} Im(sgn \overline{u}∇u). Then we have

sgn \overline{u}∇v_{n} = (φ_{n} + iψ_{n}) + \frac{p−2}{2}χ_{n}^{c}φ_{n} = (\frac{p}{2}χ_{n}^{c} + χ_{n})φ_{n} + iψ_{n}.

In the same way, with ρ_{n} := (p − 1)χ_{n}^{c} + χ_{n}, we have

∇\overline{w}_{n} = u_{n}^{2}(∇\overline{u} + (p − 2)χ_{n}^{c} sgn \overline{u}∇|u|) = u_{n} sgn \overline{u}(ρ_{n}φ_{n} − iψ_{n}).

Now we compute the different terms occurring in τ(u, w_{n}) and τ_{p}(v_{n}) separately.

a∇u · ∇\overline{w}_{n} = a(u_{n} sgn \overline{u}∇u) · (ρ_{n}φ_{n} − iψ_{n}) = a(φ_{n} + iψ_{n})(ρ_{n}φ_{n} − iψ_{n}), (3.15)
\begin{matrix}a∇v_{n} · ∇\overline{v}_{n} & = a(sgn \overline{u}∇v_{n}) · (sgn u∇\overline{v}_{n}) \\ = (\frac{p^{2}}{4} χ_{n}^{c} + χ_{n})a_{s}φ_{n} · φ_{n} + a_{s}ψ_{n} · ψ_{n} + i(a − a_{s})ψ_{n} · (pχ_{n}^{c} + 2χ_{n})φ_{n}. \end{matrix} (3.16)

Therefore Re a∇u·∇\overline{w}_{n} = ((p−1)χ_{n}^{c}+χ_{n})a_{s}φ_{n}·φ_{n}+a_{s}ψ_{n}·ψ_{n}. Noting \frac{4}{pp′} \frac{p^{2}}{4} = p−1 we obtain

Re τ_{a}(u, w_{n}) = \frac{4}{pp′} Re τ_{a}(v_{n}) + (1 − \frac{4}{pp′})⟨χ_{n}a_{s}φ_{n} · φ_{n} + a_{s}ψ_{n} · ψ_{n}⟩.

3.3. WEIGHTED ESTIMATES FOR SECOND ORDER OPERATORS 73

For the first order terms we compute

\overline{v}_{n}∇v_{n} = |v_{n}|((\frac{p}{2}χ_{n}^{c} + χ_{n})φ_{n} + iψ_{n}),
\overline{w}_{n}∇u = |v_{n}|u_{n} sgn \overline{u}∇u = |v_{n}|(φ_{n} + iψ_{n}),  (3.17)
u∇\overline{w}_{n} = |v_{n}|(ρ_{n}φ_{n} − iψ_{n}).

Thus, Re \overline{v}_{n}∇v_{n} = Re v_{n}∇\overline{v}_{n} = (\frac{p}{2}χ_{n}^{c} + χ_{n})|v_{n}|φ_{n}. We obtain that

Re \overline{w}_{n}∇u = (χ_{n}^{c} + χ_{n})|v_{n}|φ_{n} = \frac{2}{p} Re(\overline{v}_{n}∇v_{n}) + (1 − \frac{2}{p})χ_{n}|v_{n}|φ_{n}

and, since \frac{2}{p′} \frac{p}{2} = p − 1,

Re u∇\overline{w}_{n} = ((p − 1)χ_{n}^{c} + χ_{n})|v_{n}|φ_{n} = \frac{2}{p′} Re(v_{n}∇\overline{v}_{n}) + (1 − \frac{2}{p′})χ_{n}|v_{n}|φ_{n}.

Let now ε_{p} := \frac{1}{p′} − \frac{1}{p} = 1 − \frac{2}{p} = −(1 − \frac{2}{p′}). Then ε_{p}^{2} = 1 − \frac{4}{pp′}. We get

Re τ(u, w_{n}) = Re τ_{a}(u, w_{n}) + Re⟨∇u, b_{1}w_{n}⟩ − Re⟨b_{2}u, ∇w_{n}⟩ + ⟨V u, w_{n}⟩

= τ_{p}(v_{n}) + ε_{p}^{2}⟨χ_{n}a_{s}φ_{n} · φ_{n} + a_{s}ψ_{n} · ψ_{n}⟩ + ε_{p}⟨χ_{n}(b_{1} + b_{2})|v_{n}| · φ_{n}⟩.

This implies (3.12) since ε_{p}χ_{n}|(b_{1}+b_{2})v_{n}·φ_{n}| ⩽ ε_{p}^{2}χ_{n}a_{s}φ_{n}·φ_{n}+ \frac{1}{2}χ_{n}(W_{1}+W_{2})|v_{n}|^{2}, by Euclid’s inequality.

To prove (3.13), we first compute Im τ_{a}(u, w_{n}). By (3.15),

Im(a∇u · ∇\overline{w}_{n}) = ((p − 1)χ_{n}^{c} + χ_{n})aψ_{n} · φ_{n} − aφ_{n} · ψ_{n}
= (p − 2)χ_{n}^{c}a_{s}ψ_{n} · φ_{n} + (pχ_{n}^{c} + 2χ_{n})(a − a_{s})ψ_{n} · φ_{n}.

The second term on the right hand side equals Im(a∇v_{n} · ∇\overline{v}_{n}), by (3.16). The first term we estimate, using Euclid’s inequality and (3.16), as follows:

|(p − 2)χ_{n}^{c}a_{s}ψ_{n} · φ_{n}| ⩽ |p − 2|χ_{n}^{c}(\frac{p}{4}a_{s}φ_{n} · φ_{n} + \frac{1}{p}a_{s}ψ_{n} · ψ_{n})

= |1 − \frac{2}{p}|χ_{n}^{c}(\frac{p^{2}}{4} a_{s}φ_{n} · φ_{n} + a_{s}ψ_{n} · ψ_{n}) ⩽ |\frac{1}{p} − \frac{1}{p′} | Re(a∇v_{n} · ∇\overline{v}_{n}).

For the first order terms we have, by (3.17),

Im(⟨∇u, b_{1}w_{n}⟩ − ⟨b_{2}u, ∇w_{n}⟩) = ⟨(b_{1} + b_{2})|v_{n}|, ψ_{n}⟩ = − Im⟨(b_{1} + b_{2})v_{n}, ∇v_{n}⟩.

Thus, inequality (3.13) follows.

3.3 Weighted estimates for second order elliptic differential operators

In this section we prove Theorems 3.8 and 3.10. In order to apply the abstract results of Section 2.1, we need to show appropriate weighted estimates for the semigroups T_{p} constructed in Theorem 3.2. Recall that the semigroups T_{p} are associated with the form τ which is defined in (3.1). We will establish estimates on the ‘twisted semigroups’ ρT_{p}ρ^{−1}, where ρ is a weight function, by studying the ‘twisted form’ τ_{ρ} which is formally defined by τ_{ρ}(u, v) = τ(ρ^{−1}u, ρv). We point out that it is a nontrivial technical problem to establish relationship between τ_{ρ} and ρT_{p}ρ^{−1} (see, e.g., [Sem97; Prop. 3.4]). A comprehensive solution is given in Theorem 3.15 below.

Throughout this section we assume that (a) and (bV) are fulfilled. Let τ_{a}, τ, τ_{p} (1 ⩽ p < ∞) be the forms defined in Section 3.1. Recall that

I = {p ∈ [1, ∞); ω_{p} = inf{ω ∈ ℝ; τ_{p} ⩾ −ω} < ∞}.

Let ρ: Ω → (0, ∞) be locally Lipschitz continuous, Φ_{ρ} := ρ^{−1}∇ρ = ∇ ln ρ, p ∈ ^{∘} I. Assume that

a_{s}Φ_{ρ} · Φ_{ρ} = ρ^{−2}a_{s}∇ρ · ∇ρ ⩽ δτ_{p} + c_{ρ}  (3.18)

for some δ ⩾ 0, c_{ρ} ∈ ℝ. Then we can define a form τ_{ρ} by

τ_{ρ}(u, v) := τ(u, v) + ⟨∇u, a^{⊤}Φ_{ρ}v⟩ − ⟨aΦ_{ρ}u,∇v⟩ − ⟨[a_{s}Φ_{ρ}·Φ_{ρ} + (b_{1} + b_{2})·Φ_{ρ}]u, v⟩

on D(τ_{ρ}) := D(τ), due to the following observations. Firstly, by Euclid’s inequality, |b_{j} · Φ_{ρ}| ⩽ \frac{1}{4}a_{s}Φ_{ρ} · Φ_{ρ} + W_{j} for j = 1, 2. Secondly, by (a) and by (1.7) from Lemma 1.29,

(aΦ_{ρ})^{⊤}a_{s}^{−1}(aΦ_{ρ}) = Φ_{ρ}^{⊤}a^{⊤}a_{s}^{−1}aΦ_{ρ} ⩽ (1 + α^{2})a_{s}Φ_{ρ} · Φ_{ρ},  (3.19)

and in the same way (a^{⊤}Φ_{ρ})^{⊤}a_{s}^{−1}(a^{⊤}Φ_{ρ}) ⩽ (1 + α^{2})a_{s}Φ_{ρ} · Φ_{ρ}. In particular, the form τ_{ρ} is of the same type as the form τ, with new lower order coefficients

b̃_{1} = b_{1} + a^{⊤}Φ_{ρ}, b̃_{2} = b_{2} + aΦ_{ρ}, Ṽ = V − a_{s}Φ_{ρ} · Φ_{ρ} − (b_{1} + b_{2}) · Φ_{ρ}

satisfying assumption (bV).

By a straightforward computation we obtain, using the product rule,

τ_{ρ}(u, v) = τ(ρ^{−1}u, ρv) (u, v ∈ D(τ_{ρ}) such that ρ^{−1}u, ρv ∈ D(τ)).

Theorem 3.15. Assume that (a) and (bV) hold. Let p ∈ ^{∘} I, ε ∈ (0, 1). Then there exist δ > 0, k ⩾ 1, ω̃ ∈ ℝ such that, for all locally Lipschitz continuous weights ρ: Ω → (0, ∞) satisfying

a_{s}Φ_{ρ} · Φ_{ρ} ⩽ δτ_{p} + c_{ρ},
⟨(b_{1} + b_{2}) · Φ_{ρ}, u^{2}⟩ ⩽ \frac{ε}{2}τ_{p}(u) + c_{ρ}||u||_{2}^{2} (0 ⩽ u ∈ D(τ))

for some c_{ρ} ∈ ℝ, the following assertions hold:

(a) The form τ_{ρ} is associated with a positive C_{0}-semigroup T_{ρ,p}(t) = e^{−tA_{ρ,p}} on L_{p}, and

⟨A_{ρ,p}u, u|u|^{p−2}⟩ ⩾ (1 − ε)\overline{τ_{p}}(|u|^{\frac{p}{2}} sgn u) − (ω̃ + kc_{ρ})||u||_{p}^{p} (u ∈ D(A_{ρ,p})).

(b) If ρ ∧ n is a multiplication operator on D(τ) for all n ∈ ℕ then

T_{ρ,p}(t)f = ρT_{p}(t)ρ^{−1}f (f ∈ L_{∞,c}, t ⩾ 0),

where T_{p} is the positive C_{0}-semigroup on L_{p} associated with τ.

Remark 3.16. If ρ(x) = ρ_{ξ}(x) = e^{ξx} for some ξ ∈ ℝ^{N} then Φ_{ρ_{ξ}} = ξ. Thus, if a ∈ L_{∞} then (3.18) is fulfilled with δ = 0, c_{ρ} = ||a_{s}ξ · ξ||_{∞} ⩽ ||a_{s}||_{∞}|ξ|^{2}. If condition (ii) of Theorem 3.8 holds then by Proposition 3.1(c) we obtain: for all p ∈ ^{∘} I there exists C_{p} ⩾ 0 such that |⟨(b_{1} + b_{2})u^{2}⟩| ⩽ C_{p}||u||_{τ_{p}}||u||_{2} for all 0 ⩽ u ∈ D(τ). Therefore,

|⟨(b_{1} + b_{2}) · ξ, u^{2}⟩| ⩽ |ξ| · |⟨(b_{1} + b_{2})u^{2}⟩| ⩽ ||u||_{τ_{p}} · C_{p}|ξ|||u||_{2}

⩽ \frac{1}{4}||u||_{τ_{p}}^{2} + C_{p}^{2}|ξ|^{2}||u||_{2}^{2} = \frac{1}{4}τ_{p}(u) + (\frac{1}{4}(ω_{p} + 1) + C_{p}^{2}|ξ|^{2})||u||_{2}^{2}.

Let n ∈ ℕ, φ_{n} := ρ_{ξ} ∧ n. Then φ_{n} ∈ W_{∞}^{1} and a_{s}∇φ_{n} · ∇φ_{n} ⩽ ||a_{s}||_{∞}||∇φ_{n}||_{∞}^{2} < ∞. Thus, if (BC) holds then φ_{n} is a multiplication operator on D(τ), and by Theorem 3.15 we obtain the following.

Let p ∈ ^{∘} I. Then there exist ν_{p} > 0, ω̃_{p} ∈ ℝ such that for all ξ ∈ ℝ^{N}, the form τ_{ρ_{ξ}} is associated with a positive C_{0}-semigroup T_{ξ,p} on L_{p} satisfying

||T_{ξ,p}(t)|| ⩽ e^{ω̃_{p}t+ν_{p}|ξ|^{2}t}, T_{ξ,p}(t) = e^{ξx}T_{p}(t)e^{−ξx} on L_{∞,c} (t ⩾ 0).

Proof of Theorem 3.15. (a) In order to apply Theorem 3.2 we need to introduce the symmetric form τ_{ρ,p} defined by

τ_{ρ,p}(u) := Re τ_{a}(u) + \frac{2}{p}⟨∇|u|, b̃_{1}|u|⟩ − \frac{2}{p′} ⟨b̃_{2}|u|, ∇|u|⟩ + ⟨Ṽ |u|^{2}⟩

= τ_{p}(u) + ⟨(\frac{2}{p}a^{⊤} − \frac{2}{p′} a)Φ_{ρ}|u|, ∇|u|⟩ − ⟨[a_{s}Φ_{ρ} · Φ_{ρ} + (b_{1} + b_{2}) · Φ_{ρ}]|u|^{2}⟩

on D(τ_{ρ,p}) := D(τ_{ρ}). By Euclid’s inequality we obtain, using (3.19),

τ_{ρ,p}(u) ⩾ τ_{p}(u) − λ(α^{2} + 1)τ_{a}(|u|) − ⟨[(1 + \frac{1}{λ})a_{s}Φ_{ρ} · Φ_{ρ} + (b_{1} + b_{2}) · Φ_{ρ}]|u|^{2}⟩

for all λ > 0, u ∈ D(τ). By Proposition 3.1(b) we can choose λ > 0, ω̃ ∈ ℝ such that

λ(α^{2} + 1) Re τ_{a} ⩽ \frac{ε}{4}τ_{p} + ω̃.

Using the assumption on (b_{1} + b_{2}) · Φ_{ρ}, we thus obtain

τ_{ρ,p} ⩾ (1 − \frac{ε}{4} − \frac{ε}{2})τ_{p} − ω̃ − c_{ρ} − (1 + \frac{1}{λ})a_{s}Φ_{ρ} · Φ_{ρ}.

Now choose δ = \frac{ε}{4}(1 + \frac{1}{λ})^{−1} and k = 2 + \frac{1}{λ}. Then τ_{ρ,p} ⩾ (1 − ε)τ_{p} − (ω̃ + kc_{ρ}) (⩾ −(1 − ε)ω_{p} − ω̃ − kc_{ρ}) by the assumption on a_{s}Φ_{ρ} · Φ_{ρ}. An application of Theorem 3.2 completes the proof of (a).

(b) Let U_{ρ} := (α^{2} + 1)a_{s}Φ_{ρ} · Φ_{ρ} + W_{1} + W_{2} + |V |. Then U := 5U_{ρ} is τ_{a}-regular by (bV), the assumption on a_{s}Φ_{ρ} ·Φ_{ρ}, and Lemma 1.24(b). Making use of (3.19), (3.2) and Euclid’s inequality, we obtain that

Re τ_{ρ} ⩾ \frac{1}{4} Re τ_{a} − 4U_{ρ}  (3.20)

and that τ + U, τ_{ρ} + U are densely defined closed sectorial forms, with domains D(τ_{a} + U_{ρ}). For m ∈ ℕ let U_{m} := (U − m)^{+}, and A_{m}, A_{ρ,m} the m-sectorial operators associated with τ + U_{m}, τ_{ρ} + U_{m}, respectively. Due to Corollary 3.4(b), part (b) will follow by passing to the limit in

e^{−tA_{ρ,m}}f = ρe^{−tA_{m}}ρ^{−1}f (f ∈ L_{∞,c}, t ⩾ 0).

The latter formula in turn is equivalent to

(λ + A_{ρ,m})^{−1}f = ρ(λ + A_{m})^{−1}ρ^{−1}f (m ∈ ℕ, λ > m, f ∈ L_{∞,c}).  (3.21)

Let m, λ, f be given. First we show that (3.21) holds if ρ satisfies the condition

ρv ∈ Q := D(τ_{a} + U_{ρ}) for all v ∈ D := (λ + A_{m})^{−1}L_{∞,c}.  (3.22)

Then u := ρ(λ + A_{m})^{−1}ρ^{−1}f ∈ Q since ρ^{−1}f ∈ L_{∞,c}. Moreover, ρ^{−1}u ∈ D(A_{m}) ⊆ Q. For all v ∈ D we have ρv ∈ Q and hence

(τ_{ρ} + U_{m})(u, v) = (τ + U_{m})(ρ^{−1}u, ρv) = ⟨A_{m}ρ^{−1}u, ρv⟩.

Observe that D is dense in D(A_{m}) and hence dense in Q = D(τ_{ρ} + U_{m}). Thus we obtain that u ∈ D(A_{ρ,m}) and A_{ρ,m}u = ρA_{m}ρ^{−1}u. Therefore (λ + A_{ρ,m})u = f, i.e., (3.21) holds.

It remains to show that the assumption of (b) implies (3.22). Let g ∈ L_{∞,c}, v := (λ+A_{m})^{−1}g (∈ Q), ρ_{n} := ρ∧n (n ∈ ℕ). Then ρ_{n}v ∈ Q for all n ∈ ℕ, by the assumption on ρ. In particular, ρ_{n} satisfies condition (3.22). Thus, (3.21) holds with ρ_{n} in place of ρ, and we obtain

(λ + A_{ρ_{n},m})^{−1}(ρg) = ρ_{n}(λ + A_{m})^{−1}ρ_{n}^{−1}(ρg) = ρ_{n}v

for n ∈ ℕ so large that ρ_{n}^{−1}ρg = g. Moreover, by (3.20) we can estimate

Re τ_{ρ_{n}} + U_{m} ⩾ \frac{1}{4} Re τ_{a} − 4U_{ρ_{n}} + U_{m} ⩾ \frac{1}{4} Re τ_{a} + U_{ρ} − m (⩾ −m).

This implies ||(λ + A_{ρ_{n},m})^{−1}|| ⩽ \frac{1}{λ−m} and (since λ > m)

(\frac{1}{4} Re τ_{a} + U_{ρ})(ρ_{n}v) ⩽ Re(τ_{ρ_{n}} + U_{m} + λ)(ρ_{n}v) = Re⟨ρg, ρ_{n}v⟩ ⩽ \frac{1}{λ − m}||ρg||_{2}^{2}.

Therefore, (ρ_{n}v) is a bounded sequence in Q. Moreover, (|ρ_{n}v|) is pointwise increasing, and ρ_{n}v → ρv a.e. as n → ∞. Hence ρv ∈ L_{2} by monotone convergence, and ρ_{n}v → ρv in L_{2} by dominated convergence. We conclude that ρv ∈ Q, i.e., ρ satisfies (3.22).

The proof of Theorem 3.8 is based on Theorem 3.15 and the following consequence of Corollaries 2.2 and 2.5.

Proposition 3.17. Let 1 ⩽ p < ∞, T a C_{0}-semigroup on L_{p}. Assume that there exist q > p, C ⩾ 1, ω ∈ ℝ, ν ⩾ 0, m > 1 such that

||e^{ξx}T (t)e^{−ξx}||_{p→p} ⩽ Ce^{ωt+ν|ξ|^{m}t},
||T (t)||_{p→q} ⩽ Ct^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q})}e^{ωt},

for all t > 0, ξ ∈ ℝ^{N}. Then T extrapolates to a family of consistent C_{0}-semi-groups T_{r}(t) = e^{−tA_{r}} on L_{r}, r ∈ [p, q), and the angle of analyticity of T_{r} and the spectrum σ(A_{r}) do not depend on r ∈ [p, q).

If, in addition, T is L_{1}-contractive then the same holds with [1, q) in place of [p, q).

Proof. Without restriction ω = 0. Let θ ∈ (0, 1) and define q_{θ} by \frac{1}{q_{θ}} = \frac{1−θ}{q} + \frac{θ}{p}. Then (1 − θ)(\frac{1}{p} − \frac{1}{q}) = \frac{1}{p} − \frac{1}{q_{θ}}. By Stein interpolation we obtain from the assumptions that

||e^{θξx}T (t)e^{−θξx}||_{p→q_{θ}} ⩽ Ct^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q_{θ}})}e^{θν|ξ|^{m}t} (t > 0, ξ ∈ ℝ^{N}).

Let d be the supremum metric on ℝ^{N}, ρ_{γ,y} := e^{−γd(·,y)} for all γ ∈ ℝ, y ∈ ℝ^{N}, and || · ||_{p→q,γ} the corresponding weighted operator norm. Replace ξ by \frac{ξ}{θ} in the above estimate. Then Proposition 2.8, applied with ψ(x) = x, yields

||T (t)||_{p→q_{θ},γ} ⩽ 2NCt^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q_{θ}})}e^{θ^{1−m}νγ^{m}t} (t, γ > 0).

Now we are in a position to apply Corollaries 2.2 and 2.5, and we obtain the first set of assertions.

The last assertion follows from the estimate

||T (t)||_{1→q_{θ},γ} ⩽ C_{1}t^{− \frac{N}{m} (1− \frac{1}{q_{θ}})}e^{ν_{1}γ^{m}t} (t, γ > 0),

for some C_{1}, ν_{1} > 0, which in turn is a consequence of the next proposition.

Proposition 3.18. Let T be a contractive C_{0}-semigroup on L_{1} and ρ: Ω → (0, ∞) a weight function. Assume that

||ρ^{γ}T (t)ρ^{−γ}||_{p→q} ⩽ Ct^{−α( \frac{1}{p} − \frac{1}{q})}e^{νγ^{m}t} (t, γ > 0)

for some 1 < p < q, C, α, ν > 0, m > 1. Then there exist C_{1}, ν_{1} > 0 (not depending on T, ρ) such that

||ρ^{γ}T (t)ρ^{−γ}||_{1→q} ⩽ C_{1}t^{−α(1− \frac{1}{q})}e^{ν_{1}γ^{m}t} (t, γ > 0).

Proof. For 0 < θ ⩽ 1 let p_{θ} := (\frac{θ}{p} + \frac{1−θ}{1})^{−1}, q_{θ} := (\frac{θ}{q} + \frac{1−θ}{1})^{−1}. By Stein interpolation, the assumptions imply that

||ρ^{θγ}T (t)ρ^{−θγ}||_{p_{θ}→q_{θ}} ⩽ C^{θ}t^{−θα( \frac{1}{p} − \frac{1}{q})}e^{θνγ^{m}t} (t, γ > 0).  (3.23)

Let t, γ > 0, determine θ ∈ (0, 1) such that q_{θ} = p, and let θ_{k} := θ^{k}, t_{k} := θ_{k}^{m}t (k ∈ ℕ_{0}) and β := α(\frac{1}{p} − \frac{1}{q}). Then p_{θ_{k}} = q_{θ_{k+1}} (k ∈ ℕ_{0}), and (3.23) yields

||ρ^{γ}T (t_{k})ρ^{−γ}||_{q_{θ_{k+1}}→q_{θ_{k}}} ⩽ C^{θ_{k}}t_{k}^{−θ_{k}β}e^{θ_{k}ν(γ/θ_{k})^{m}t_{k}} = C^{θ_{k}}(θ^{mk}t)^{−θ_{k}β}e^{θ_{k}νγ^{m}t}

for all k ∈ ℕ_{0}. We use this as a starting point for a Moser type iteration: let f ∈ L_{∞,c}. Since q = q_{θ_{0}} we obtain by Fatou’s lemma that

||ρ^{γ}T (\frac{t}{1−θ^{m}})ρ^{−γ}f||_{q} ⩽ lim inf_{n→∞} ||ρ^{γ}T (∑_{k=0}^{n} t_{k})ρ^{−γ}f||_{q_{θ_{0}}}

⩽ lim inf_{n→∞} ∏_{k=0}^{n}(C^{θ_{k}}θ^{−mβkθ_{k}}t^{−θ_{k}β}e^{θ_{k}νγ^{m}t}) · ||f||_{q_{θ_{n+1}}}.

Let r := ∑_{k=0}^{∞} θ_{k} = \frac{1}{1−θ} and s := ∑_{k=0}^{∞} kθ_{k}(= \frac{θ}{(1−θ)^{2}} ). By the choice of θ we have ∑_{k=0}^{∞} θ_{k}β = \frac{α}{1−θ}(\frac{1}{q_{θ}} − \frac{1}{q}) = α(1 − \frac{1}{q}). We conclude that

||ρ^{γ}T (\frac{t}{1−θ^{m}})ρ^{−γ}f||_{q} ⩽ C^{r}θ^{−mβs}t^{−α(1− \frac{1}{q})}e^{rνγ^{m}t}||f||_{1}.

This yields the assertion with C_{1} = C^{r}θ^{−mβs}(1 − θ^{m})^{−α(1− \frac{1}{q})} and ν_{1} = (1 − θ^{m})rν.

Proof of Theorem 3.8. Recall that assumptions (i) and (ii) of the theorem are fulfilled for all p ∈ ^{∘} I (see Remark 3.9(a)). Let p ∈ ^{∘} I = (p_{−}, p_{+}), T_{p}(t) = e^{−tA_{p}} the positive C_{0}-semigroup on L_{p} associated with the form τ. We are going to apply Proposition 3.17 with q = \frac{N}{N−2}p and m = 2. By Remark 3.16 we have

||e^{ξx}T_{p}(t)e^{−ξx}||_{p→p} ⩽ e^{ω̃_{p}t+ν_{p}|ξ|^{2}t} (t > 0, ξ ∈ ℝ^{N}),

i.e., the first estimate assumed in Proposition 3.17 holds.

In order to show the second estimate, let 0 ⩽ f ∈ L_{p}, t > 0, u := T_{p}(t)f. Then u ∈ D(A_{p}) since T_{p} is analytic, and u ⩾ 0. By Theorem 3.2(b) and assumption (i) of the theorem we have

⟨A_{p}u, u^{p−1}⟩ ⩾ \overline{τ_{p}}(u^{\frac{p}{2}} ) ⩾ ε_{p}||u^{\frac{p}{2}} ||2_{\frac{2N}{N−2}} − c_{p}||u^{\frac{p}{2}} ||_{2}^{2}.

Without restriction assume c_{p} = 0. Then T_{p} is contractive. Since (p − 1)p′ = p we obtain, using Hölder’s inequality and the analyticity of T_{p},

⟨A_{p}u, u^{p−1}⟩ ⩽ ||A_{p}T_{p}(t)f||_{p}||(T_{p}(t)f)^{p−1}||_{p′} ⩽ \frac{c}{t}||f||_{p}||T_{p}(t)f||_{p}^{p−1} ⩽ \frac{c}{t}||f||_{p}^{p}

for some c > 0 (not depending on t, f). Combining the above two estimates, we arrive at

ε_{p}||T_{p}(t)f||_{\frac{N}{N−2} p}^{p} ⩽ \frac{c}{t}||f||_{p}^{p},

so that, by the positivity of T_{p},

||T_{p}(t)||_{p→ \frac{N}{N−2} p} ⩽ Ct^{− \frac{1}{p}} = Ct^{− \frac{N}{2} (p^{−1}−( \frac{N}{N−2} p)^{−1})} (t > 0).  (3.24)

By Proposition 3.17 we conclude that T_{p} extrapolates to an analytic semigroup T_{r}(t) = e^{−tA_{r}} on L_{r}, for all r ∈ [p, \frac{N}{N−2}p), and that the angle of analyticity of T_{r} and the spectrum σ(A_{r}) are r-independent. In case 1 ∈ I, the same holds with [1, \frac{N}{N−2}p) in place of [p, \frac{N}{N−2}p). Moreover, the semigroups T_{r} are associated with τ, by Proposition 1.22(a). Thus, in case 1 ∈ I the proof is complete while otherwise we obtain the assertions of the theorem only with (p_{−}, p_{max}) in place of (p_{min}, p_{max}).

To complete the proof in the case 1 ∉ I, we apply the above to the form τ^{∗} in place of τ. The form τ^{∗} is of the same type as the form τ, with coefficients ã = a^{⊤}, b̃_{1} = −b_{2}, b̃_{2} = −b_{1}, Ṽ = V. It is easy to see that (τ^{∗})_{p} = τ_{p′} for all p ∈ (1, ∞). Therefore, τ^{∗} is associated with an analytic semigroup T̂_{r}(t) = e^{−tÂ_{r}} on L_{r}, for all r ∈ (p′_{+}, p′_{min}), and the angle of analyticity of T̂_{r} and the spectrum σ(Â_{r}) are r-independent. By Proposition 1.22(b) we conclude that τ is associated with the positive C_{0}-semigroup T̂_{r′}^{∗}(t) = e^{−tA_{r′}^{∗}} on L_{r}, for all r ∈ (p_{min}, p_{+}). It now remains to note the following: T̂_{r′}^{∗} is analytic of angle θ ∈ (0, \frac{π}{2}] if and only if T̂_{r′} is analytic

of angle θ, and σ(Â_{r′}^{∗}) = \overline{σ(Â_{r′})} (= σ(Â_{r′}) since T̂_{r′} is a real semigroup).

In the proof of Theorem 3.10, it will be a bit more difficult to prove p→qsmoothing for the semigroup since the assumptions do not ensure analyticity. In fact, it is easier to show p→q-smoothing for the resolvent and use Theorem 2.26 instead of Theorem 2.4 to obtain L_{p}-spectral independence. We will use the latter approach for the case p > 1. As above, the case p = 1 requires additional expense. In order to show a weighted 1→q-estimate, we are going to use Proposition 3.18. For this reason we need to show p→q-smoothing for the semigroup. Alternatively, we could prove an analogue of Proposition 3.18 for resolvents, cf. [Sem97; proof of Prop. 4.2].

Both p→q-smoothing for the semigroup and for the resolvent are consequences of an improved accretivity estimate. We present this well-known technique in a separate lemma. As a preparation we need the following fact.

Lemma 3.19. Let (Ω, μ) be a measure space, p ∈ (1, ∞), ∅ ≠ J ⊆ ℝ an interval. Let u: J → L_{p}(μ) be differentiable, u(t) ⩾ 0 for all t ∈ J. Then the function u^{p}: J → L_{1}(μ) is differentiable, with

(u^{p})′ = pu^{p−1}u′.

Proof. We first show that for x, y ⩾ 0 we have

\begin{matrix}0 & ⩽ x^{p} − y^{p} − py^{p−1}(x − y) ⩽ |x − y|^{p} (1 < p ⩽ 2), \\ 0 & ⩽ x^{p} − y^{p} − py^{p−1}(x − y) ⩽ \frac{1}{2}p(p − 1)(x ∨ y)^{p−2}|x − y|^{2} (p ⩾ 2). \end{matrix} (3.25)

For the proof let f: [0, ∞) → [0, ∞), f(x) := x^{p}. Then f is convex, and x ↦ y^{p} + py^{p−1}(x − y) is the tangent to f in (y, y^{p}), for all y ⩾ 0. This shows the left hand sides of the inequalities. For the proof of the right hand side for p ⩾ 2, just apply Taylor’s theorem with Lagrangian remainder to f. It remains to show that g_{y}(x) := x^{p} − y^{p} − py^{p−1}(x − y) − |x − y|^{p} ⩽ 0 in case 1 < p ⩽ 2, for all x, y ⩾ 0. We have g_{y}(y) = 0, and \frac{1}{p}g′_{y}(x) = x^{p−1} − y^{p−1} − |x − y|^{p−1} sgn(x − y). The subadditivity of the function x ↦ x^{p−1} shows that g′_{y}(x) ⩾ 0 for x ⩽ y and g′_{y}(x) ⩽ 0 for x ⩾ y. This completes the proof of (3.25).

Let now s, t ∈ J, s ≠ t, and assume that p ⩽ 2. Applying (3.25) to x = u(s) and y = u(t) we obtain, after division by |s − t|,

|\frac{u(s)^{p} − u(t)^{p}}{s − t} − pu(t)^{p−1} \frac{u(s) − u(t)}{s − t}| ⩽ |s − t|^{p−1} |\frac{u(s) − u(t)}{s − t}|^{p} .

Letting s → t yields the assertion for 1 < p ⩽ 2. The case p > 2 is proved similarly (use the fact that ||v^{p−2}w^{2}||_{1} ⩽ ||v||_{p}^{p−2}||w||_{p}^{2} for all v, w ∈ L_{p}(μ)).

If we allow u to be complex-valued in the above lemma then we obtain (|u|^{p})′ = p|u|^{p−1} Re(u′ sgn u). This follows from |u|′(t) = Re(u′(t) sgn u(t)) + |u′(t)|χ_{[u(t)=0]} (cf. [Nag86; Prop. B-II.2.3, Example C-II.2.3]) and the chain rule.

Lemma 3.20. Let (Ω, μ) be a measure space, p ∈ (1, ∞). Let T (t) = e^{−tA} be a positive C_{0}-semigroup on L_{p}(μ) satisfying

⟨Au, u^{p−1}⟩ ⩾ ε||u||_{rp}^{p} (0 ⩽ u ∈ D(A))

for some r > 1, ε > 0.

(a) Then A is m-accretive, and ||(λ + A)^{−1}||_{(rp′)′→rp} ⩽ \frac{1}{ε} for all λ > 0.

(b) Assume that ||T (t)||_{rp→rp} ⩽ C (t ⩾ 0) for some C ⩾ 1. Then ||T (t)||_{p→rp} ⩽ C(εpt)^{− \frac{1}{p}} for all t > 0.

Proof. (a) The semigroup T is contractive since it is positive and ⟨Au, u^{p−1}⟩ ⩾ 0 for all 0 ⩽ u ∈ D(A). Thus, A is m-accretive. Let now λ > 0, 0 ⩽ f ∈ L_{p}(μ) ∩ L_{(rp′)′}(μ), and u := (λ + A)^{−1}f. Then u ⩾ 0 and hence

⟨f, u^{p−1}⟩ = ⟨(λ + A)u, u^{p−1}⟩ ⩾ ε||u||_{rp}^{p}.

We estimate the left hand side, using Hölder’s inequality and noting (p−1)p′ = p,

⟨f, u^{p−1}⟩ ⩽ ||f||_{(rp′)′}||u^{p−1}||_{rp′} = ||f||_{(rp′)′}||u||_{rp}^{p−1}.

Combining the above two estimates, we obtain ε||u||_{rp} ⩽ ||f||_{(rp′)′}. By the positivity of T, this proves (a).

(b) Let 0 ⩽ f ∈ D(A), and u_{t} := T (t)f for all t ⩾ 0. Then −\frac{d}{ds}u_{s} = Au_{s} for all s ⩾ 0 and hence, by Lemma 3.19,

−\frac{d}{ds}||u_{s}||_{p}^{p} = −\frac{d}{ds}⟨u_{s}^{p}⟩ = p⟨Au_{s}, u_{s}^{p−1}⟩ ⩾ εp||u_{s}||_{rp}^{p}.

By the assumption of (b), ||u_{t}||_{rp} ⩽ C||u_{s}||_{rp} for all 0 ⩽ s ⩽ t, so we obtain

εpt||u_{t}||_{rp}^{p} ⩽ ∫_{0}^{t} εpC^{p}||u_{s}||_{rp}^{p} ds ⩽ C^{p} ∫_{0}^{t}(−\frac{d}{ds}||u_{s}||_{p}^{p}) ds ⩽ C^{p}||f||_{p}^{p} (t > 0),

noting u_{0} = f and ||u_{t}||_{p}^{p} ⩾ 0. Thus, ||T (t)f||_{rp} ⩽ C(εpt)^{− \frac{1}{p}} ||f||_{p} for all t > 0, 0 ⩽ f ∈ D(A). The set of those f is dense in {f ∈ L_{p}(μ); f ⩾ 0}, so the proof is complete by the positivity of T.

If we do not assume the semigroup T to be positive then the assertions of the above lemma still hold if we replace the assumption by

Re⟨Au, u|u|^{p−2}⟩ ⩾ ε||u||_{rp}^{p} (u ∈ D(A)).

We will apply Theorem 2.26 in the proof of Theorem 3.10 via the following result.

Proposition 3.21. ([LiVo00; Thm. 1]) Given 1 ⩽ p < q < ∞, let A_{p}, A_{q} be closed operators in L_{p}, L_{q}, respectively. Assume that there exist p_{0} ⩽ p, q_{0} ⩾ q, γ > 0, C < ∞, λ_{0} ∈ ρ(A_{p}) ∩ ρ(A_{q}), and an L^{1}-regular function ψ: ℝ^{N} → ℝ^{N} such that (λ_{0} − A_{p})^{−1}, (λ_{0} − A_{q})^{−1} are consistent and

||e^{ξψ}(λ_{0} − A_{p})^{−1}e^{−ξψ}||_{p_{0}→q_{0}} ⩽ C for all ξ ∈ ℝ^{N}, |ξ| ⩽ γ.

Then σ(A_{p}) = σ(A_{q}), and (λ − A_{p})^{−1}, (λ − A_{q})^{−1} are consistent for all λ ∈ ρ(A_{p}) = ρ(A_{q}).

Proof. Let μ be the Lebesgue measure on ℝ^{N} and d the semi-metric on ℝ^{N} defined by d(x, y) := |ψ(x) − ψ(y)|_{∞}. For x ∈ ℝ^{N} =: M, r > 0 let v_{r}(x) := μ(B(x, r)). By the paragraph preceding Lemma 2.9 we know the following. Conditions (2.7) and (2.8) are fulfilled, and there exists C_{1} > 0 such that, with the weighted operator norm corresponding to the semi-metric d, we have

||v_{1}^{α}(λ_{0} − A_{p})^{−1}||_{p_{0}→q_{0},γ} ⩽ C_{1} (0 ⩽ α ⩽ 1).

By Remark 2.28 we conclude that

||v_{1}^{\frac{1}{p} − \frac{1}{q}} (λ_{0} − A_{p})^{−1}||_{p→q, \frac{γ}{2}} ⩽ K_{γ}C_{1}.

Now Theorem 2.26 implies that (λ_{0} − A_{p})^{−1}, (λ_{0} − A_{q})^{−1} have equal spectra and consistent resolvents. The spectral mapping theorem for the resolvent yields σ(A_{p}) = σ(A_{q}), and the consistency of the resolvents follows from the identity (λ− A)^{−1} = (λ_{0}−λ)^{−1}(λ_{0}−A)^{−1}((λ_{0}−λ)^{−1}−(λ_{0}−A)^{−1})^{−1} (cf. Proposition 2.33).

Proof of Theorem 3.10. Recall from Proposition 3.1(c) that the first assumption of the theorem implies that for all p ∈ ^{∘} I there exist ε_{p} > 0, c_{p} ∈ ℝ such that

τ_{p}(u) ⩾ ε_{p}||u||_{2r}^{2} − c_{p}||u||_{2}^{2} (u ∈ D(τ)).  (3.26)

(a) For ξ ∈ ℝ^{N} let ρ_{ξ} := e^{ξψ}, and τ_{ρ_{ξ}} the corresponding form defined in the beginning of the section. Then Φ_{ρ_{ξ}} = ∇ ln ρ_{ξ} = ∇(ξψ). By Euclid’s inequality it follows that

a_{s}Φ_{ρ_{ξ}} · Φ_{ρ_{ξ}} = ∑_{j,k=1}^{N} ξ_{j}ξ_{k}a_{s}∇ψ_{j} · ∇ψ_{k} ⩽ c|ξ|^{2} ∑_{j=1}^{N} a_{s}∇ψ_{j} · ∇ψ_{j},  (3.27)

with a constant c ⩾ 1 depending only on the dimension N. Similarly,

|⟨(b_{1} + b_{2}) · Φ_{ρ_{ξ}}, u^{2}⟩| ⩽ |ξ| ∑_{j=1}^{N} |⟨(b_{1} + b_{2}) · ∇ψ_{j}, u^{2}⟩| (0 ⩽ u ∈ D(τ)).

Let p ∈ ^{∘} I, T_{p} the C_{0}-semigroup associated with τ. Let ε = \frac{1}{2} and choose δ, k, ω̃ as in Theorem 3.15. By Proposition 3.1(c) and the assumptions of (a), the above two estimates imply

a_{s}Φ_{ρ_{ξ}} · Φ_{ρ_{ξ}} ⩽ |ξ|^{2}(c_{0,p}τ_{p} + c_{1,p}),  (3.28)
⟨(b_{1} + b_{2}) · Φ_{ρ_{ξ}}, u^{2}⟩ ⩽ |ξ|(c_{0,p}τ_{p}(u) + c_{1,p}||u||_{2}^{2}) (0 ⩽ u ∈ D(τ)),

for some c_{0,p}, c_{1,p} > 0 and all ξ ∈ ℝ^{N}. Thus the assumptions of Theorem 3.15 are fulfilled for ρ = ρ_{ξ}, with c_{ρ} = 1, if |ξ| is sufficiently small.

We conclude that there exists γ > 0 such that, for all |ξ| ⩽ γ, the form τ_{ρ_{ξ}} is associated with a positive C_{0}-semigroup T_{ξ,p}(t) = e^{−tA_{ξ,p}} on L_{p}, and

⟨A_{ξ,p}u, u^{p−1}⟩ ⩾ \frac{1}{2}\overline{τ_{p}}(u^{\frac{p}{2}} ) − (ω̃ + k)||u||_{p}^{p} (0 ⩽ u ∈ D(A_{ξ,p})).

By (3.26) we obtain that there exists C_{p} ∈ ℝ such that

⟨A_{ξ,p}u, u^{p−1}⟩ ⩾ \frac{1}{2}ε_{p}||u^{\frac{p}{2}} ||_{2r}^{2} − C_{p}||u^{\frac{p}{2}} ||_{2}^{2} = \frac{ε_{p}}{2} ||u||_{rp}^{p} − C_{p}||u||_{p}^{p} (0 ⩽ u ∈ D(A_{ξ,p})).

Let n ∈ ℕ, φ_{n} := e^{ξψ} ∧ n. Then φ_{n} ∈ W_{∞}^{1}, ∇φ_{n} = e^{ξψ}χ_{[e^{ξψ}<n]}∇(ξψ) = φ_{n}χ_{[φ_{n}<n]}Φ_{ρ_{ξ}}. Hence, by (3.28),

a_{s}∇φ_{n} · ∇φ_{n} ⩽ n^{2}a_{s}Φ_{ρ_{ξ}} · Φ_{ρ_{ξ}} ⩽ c_{ξ}(Re τ_{a} + W_{1} + W_{2} + V^{+} + 1)

for some c_{ξ} > 0. Since (BC) holds, φ_{n} is a multiplication operator on D(τ) for all n ∈ ℕ, so Theorem 3.15(b) yields T_{ξ,p}(t)f = e^{ξψ}T_{p}(t)e^{−ξψ}f for all f ∈ L_{∞,c}, t ⩾ 0. Thus, (λ + A_{ξ,p})^{−1}f = e^{ξψ}(λ + A_{p})^{−1}e^{−ξψ}f for all f ∈ L_{∞,c}, λ > C_{p}.

Now, since ⟨(C_{p} + A_{ξ,p})u, u^{p−1}⟩ ⩾ \frac{ε_{p}}{2} ||u||_{rp}^{p} (0 ⩽ u ∈ D(A_{ξ,p})), Lemma 3.20(a) implies that ||e^{ξψ}(λ + A_{p})^{−1}e^{−ξψ}||_{(rp′)′→rp} ⩽ \frac{2}{ε_{p}} for all λ > C_{p}, |ξ| ⩽ γ. By Proposition 3.21 we infer that σ(A_{q}) is independent of q ∈ [(rp′)′, rp] ∩ I, for all p ∈ ^{∘} I, and that the operators A_{q} have consistent resolvents. This proves (a).

(b) Assume that 1 ∈ I; otherwise there is nothing to show. Let p ∈ ^{∘} I such that p ⩽ r and p^{2} ∈ I. In a first step, we use Theorem 3.15 to obtain a weighted p→p-estimate for the semigroup T_{p}. Then we use Proposition 3.18 to derive a weighted 1→q-estimate, for some q ∈ (1, p).

(i) For all δ > 0, u ∈ D(τ) we have

||u||_{τ_{p}}^{2−ε}||u||_{2}^{ε} = ||δu||_{τ_{p}}^{2−ε}||δ^{1− \frac{2}{ε}} u||_{2}^{ε} ⩽ ||δu||_{τ_{p}}^{2} + ||δ^{1− \frac{2}{ε}} u||_{2}^{2}.

We deduce that, for all δ > 0, u ∈ D(τ), ξ ∈ ℝ^{N},

|ξ|^{2}||u||_{τ_{p}}^{2−ε}||u||_{2}^{ε} ⩽ |ξ|^{2}(\frac{\sqrt{δ}}{|ξ|})^{2}||u||_{τ_{p}}^{2} + |ξ|^{2}(\frac{\sqrt{δ}}{|ξ|})^{2− \frac{4}{ε}} ||u||_{2}^{2} = δ||u||_{τ_{p}}^{2} + δ^{1− \frac{2}{ε}} |ξ|^{\frac{4}{ε}} ||u||_{2}^{2}.

Let ε = \frac{1}{2} and choose δ, k, ω̃ as in Theorem 3.15. By (3.27) and the assumption of (b), the above implies that

a_{s}Φ_{ρ_{ξ}} · Φ_{ρ_{ξ}} ⩽ c|ξ|^{2} ∑_{j=1}^{N} a_{s}∇ψ_{j} · ∇ψ_{j} ⩽ δτ_{p} + C(1 + |ξ|^{\frac{4}{ε}}) (ξ ∈ ℝ^{N}),

for some C ⩾ 1. In the same way,

|⟨(b_{1} + b_{2}) · Φ_{ρ_{ξ}}, u^{2}⟩| ⩽ \frac{1}{4}τ_{p}(u) + C(1 + |ξ|^{\frac{2}{ε}})||u||_{2}^{2} ⩽ \frac{1}{4}τ_{p}(u) + C(2 + |ξ|^{\frac{4}{ε}} )||u||_{2}^{2}

for all u ∈ D(τ), ξ ∈ ℝ^{N}. Thus, for all ξ ∈ ℝ^{N} the assumptions of Theorem 3.15 are fulfilled for ρ = ρ_{ξ}, with c_{ρ} = C(2 + |ξ|^{\frac{4}{ε}}). Since assumption (BC) holds we can apply Theorem 3.15, as in the proof of (a), to conclude that

||e^{ξψ}T_{p}(t)e^{−ξψ}||_{p→p} ⩽ e^{(ω̃+kC(2+|ξ|^{\frac{4}{ε}}))t} (t ⩾ 0, ξ ∈ ℝ^{N}).

(ii) Recall that p ⩽ r. Thus, ||u||_{2r}^{2} ⩾ ||u||_{2p}^{2} − ||u||_{2}^{2} for all u ∈ L_{2} ∩ L_{r}. By Theorem 3.2(b) and (3.26) we obtain that

⟨A_{p}u, u^{p−1}⟩ ⩾ \overline{τ_{p}}(u^{\frac{p}{2}} ) ⩾ ε_{p}(||u^{\frac{p}{2}} ||_{2p}^{2} − ||u^{\frac{p}{2}} ||_{2}^{2}) − c_{p}||u^{\frac{p}{2}} ||_{2}^{2} (0 ⩽ u ∈ D(A_{p})).

Let ω := (c_{p} + ε_{p}) ∨ ω_{p^{2}}. Then ⟨(ω + A_{p})u, u^{p−1}⟩ ⩾ ε_{p}||u||_{p^{2}}^{p} for all 0 ⩽ u ∈ D(A_{p}). Moreover, e^{−ωt}T_{p^{2}}(t) is a contractive semigroup, so by Lemma 3.20(b) we infer that

||T_{p}(t)||_{p→p^{2}} ⩽ (ε_{p}pt)^{− \frac{1}{p}} e^{ωt} = (ε_{p}pt)^{−p′( \frac{1}{p} − \frac{1}{p^{2}} )}e^{ωt} (t > 0).

Let now q ∈ (1, p). Then Stein interpolation between the above p→p^{2}-esti-mate and the weighted p→p-estimate obtained in step (i) yields

||e^{ξψ}T_{p}(t)e^{−ξψ}||_{p→pq} ⩽ C_{1}t^{−p′( \frac{1}{p} − \frac{1}{pq})}e^{ν_{1}(1+|ξ|^{\frac{4}{ε}})t} (t > 0, ξ ∈ ℝ^{N})

for some C_{1}, ν_{1} > 0. Finally, we apply Proposition 3.18, and once again Stein interpolation, to obtain

||e^{ξψ}T_{p}(t)e^{−ξψ}||_{1→q} ⩽ C_{2}t^{−p′(1− \frac{1}{q})}e^{ν_{2}(1+|ξ|^{\frac{4}{ε}})t} (t > 0, ξ ∈ ℝ^{N})

for some C_{2}, ν_{2} > 0. Note that the t-exponent is greater that −1 since q < p. From this we deduce that the assumptions of Proposition 3.21 are fulfilled, with p_{0} = p = 1 and q_{0} = q. Thus, A_{1} and A_{q} have equal spectra and consistent resolvents.

3.4 Sharpness of the results

In this section we show that, under some conditions additional to (a) and (bV), the interval I of quasi-contractivity obtained in Theorem 3.2 cannot be enlarged (up to possibly adding p = 1). Expressed differently, if τ is associated with a quasi-contractive C_{0}-semigroup on L_{p}, for some p ∈ (1, ∞), then τ_{p} is bounded below.

Later on, we give an example of coefficients b_{1}, b_{2}, V where the interval of existence of the semigroup obtained in Theorem 3.8 cannot be further extended. In this sense, the interval extension given in Theorem 3.8 is optimal. The contents of this section are partly due to Z. Sobol.

The following theorem is the main part of our sharpness result. It is valid under an assumption slightly weaker than (bV), namely

(bV’) the potentials W_{1}, W_{2}, |V | are τ_{a}-regular.

Under this assumption, the forms τ and τ_{p} can be defined in the same way as in Section 3.1, on D(τ) = D(τ_{p}) = D(τ_{a} + W_{1} + W_{2} + |V |).

Theorem 3.22. Let (a) and (bV’) hold, and p ∈ (1, ∞). Assume that τ ↔ T_{p} on L_{p}, with ||T_{p}(t)|| ⩽ e^{ω_{p}t} (t ⩾ 0) for some ω_{p} ∈ ℝ. In the case p > 2 (p < 2) additionally assume that sup_{t⩾0} ||(T_{p})_{U}(t)||_{∞→∞} < ∞ (sup_{t⩾0} ||(T_{p})_{U}(t)||_{1→1} < ∞) for some τ_{a}-regular potential U ⩾ 0. Then τ_{p} ⩾ −ω_{p}.

The additional assumption in the case p > 2 is in particular fulfilled in the following situation. Suppose that ⟨∇u, b_{2}u⟩ ⩽ ω||u||_{2}^{2} for all 0 ⩽ u ∈ D(τ) (e.g. b_{2} = 0). Let U := V^{−} + ω. Then τ + U ↔ (T_{p})_{U} by Corollary 3.4(a). Note that (τ_{∞} + U)(u) ⩾ 0 for all 0 ⩽ u ∈ D(τ). Thus, ||(T_{p})_{U}(t)||_{∞→∞} ⩽ 1 for all t ⩾ 0, by Remark 3.3(e).

The proof of Theorem 3.22 is based on the following lemma.

Lemma 3.23. Let (M, μ) be a measure space, 𝔥 a Dirichlet form in L_{2}(μ), D(𝔥)_{+} the set of positive elements of D(𝔥), and r ⩾ 1.

(a) Then D_{1} := {u ∈ D(𝔥)_{+} ∩ L_{∞}(μ); u^{1/r} ∈ D(𝔥)} is dense in D(𝔥)_{+}.

(b) Let 𝔥_{1} be a densely defined closed sectorial form in L_{2}(μ) fulfilling the first Beurling-Deny criterion, A the m-sectorial operator associated with 𝔥_{1}. Assume that D(𝔥_{1}) = D(𝔥), and ||e^{−tA}||_{∞→∞} ⩽ C (0 ⩽ t ⩽ 1) for some C > 0. Then D_{2} := {u^{r}; 0 ⩽ u ∈ D(A) ∩ L_{∞}(μ), Au ∈ L_{∞}(μ)} is dense in D(𝔥)_{+}.

Proof. (a) For n ∈ ℕ define φ_{n}: [0, ∞) → [0, n] by φ_{n}(s) := s ∧ (ns^{r}) ∧ n. It is easy to show that φ_{n} is Lipschitz continuous with constant r, φ_{n}^{1/r} is Lipschitz continuous, and φ_{n}(s) → s as n → ∞ (s ⩾ 0). For u ∈ D(𝔥)_{+} we conclude that φ_{n}(u) ∈ D_{1}, and from [Anc76; Prop. 11] we deduce that φ_{n}(u) → u in D(𝔥) as n → ∞.

(b) By (a), it remains to show that D_{2} is dense in D_{1} with respect to ||·||_{𝔥}. Let u ∈ D_{1} and v := u^{1/r}. Then v ∈ D(𝔥) ∩ L_{∞}(μ). By [MaRö92; Thm. I.2.13(ii)] we have v_{λ} := λ(λ + A)^{−1}v → v in D(𝔥_{1}) and thus in D(𝔥) as λ → ∞. The assumptions on 𝔥_{1} and A imply that 0 ⩽ v_{λ} ∈ D(A) ∩ L_{∞} and ||v_{λ}||_{∞} ⩽ 2C||v||_{∞} for sufficiently large λ. Moreover, we have Av_{λ} = λ(v − v_{λ}) ∈ L_{∞}. Therefore, v_{λ}^{r} ∈ D_{2}. Note that the function s ↦ s^{r} in Lipschitz continuous on [0, 2C||v||_{∞}]. Hence, by [Anc76; Théorème 10], v_{λ}^{r} → v^{r} = u in D(𝔥) as λ → ∞.

We further need the following trivial but nevertheless important fact. Let E, F be Banach spaces, and assume that there exists a vector space G such that E ↪ G, F ↪ G.

Lemma 3.24. Let A_{E}, A_{F} be closed operators in E, F, respectively. Assume that there exists λ ∈ ρ(A_{E}) ∩ ρ(A_{F}) such that (λ − A_{E})^{−1}, (λ − A_{F})^{−1} are consistent. Let u ∈ D(A_{E}) ∩ F such that A_{E}u ∈ F. Then u ∈ D(A_{F}), A_{F} u = A_{E}u.

Proof. We have (λ − A_{E})u ∈ E ∩ F. Hence

u = (λ − A_{E})^{−1}(λ − A_{E})u = (λ − A_{F})^{−1}(λ − A_{E})u.

This implies u ∈ D(A_{F}), (λ − A_{F})u = (λ − A_{E})u.

Proof of Theorem 3.22. It suffices to study the case p ⩾ 2. Then the assertion for the case p < 2 follows by an application of Proposition 1.22(b) (recall from the proof of Theorem 3.8 that τ_{p} = (τ^{∗})_{p′}). In the case p > 2 assume without restriction that U ⩾ U_{0} := W_{1}+W_{2}+2|V | (see Lemma 1.24(b)). In the case p = 2 let U := U_{0}. Then τ +U is a closed sectorial form in L_{2} (cf. Proposition 3.1(a); the factor 2 is needed since we do not assume Q(V^{−}) ⊇ D(τ_{a}) ∩ Q(W_{1} + W_{2} + V^{+})).

Let A_{p}, A_{p,U} be the generators of T_{p}, (T_{p})_{U}, respectively. By the Lumer-Phillips theorem and the assumption on T_{p} we have ⟨A_{p}u, u|u|^{p−2}⟩ ⩾ −ω_{p}||u||_{p}^{p} for all u ∈ D(A_{p}). By Lemma 1.34 we infer that

⟨A_{p,U}u, u|u|^{p−2}⟩ ⩾ ⟨(U − ω_{p})|u|^{p}⟩ (u ∈ D(A_{p,U})).  (3.29)

We have to prove τ_{p} ⩾ −ω_{p} on D(τ_{p}) = D(τ_{a} + U_{0}). Notice that τ_{p} is a bounded form on D(τ_{a} + U_{0}). Since U is (τ_{a} + U_{0})-regular, by Lemma 1.24(b), it therefore suffices to show τ_{p}(u) ⩾ −ω_{p}||u||_{2}^{2} for all u ∈ D(τ_{a} + U). Let first p = 2. Then A_{2,U} is the m-sectorial operator associated with τ + U, so

(τ_{2} + U)(u) = Re(τ + U)(u) = Re⟨A_{2,U}u, u⟩ ⩾ ⟨(U − ω_{2})u^{2}⟩ (u ∈ D(A_{2,U})), by (3.29). This shows the assertion for p = 2 since D(A_{2,U}) is dense in D(τ +U) = D(τ_{a} + U).

Let now p > 2. Let A_{U} be the m-sectorial operator in L_{2} associated with τ + U. Below we show that

τ_{p}(u^{\frac{p}{2}} ) ⩾ −ω_{p}||u^{\frac{p}{2}} ||_{2}^{2}  (3.30)

for all 0 ⩽ u ∈ D(A_{U}) ∩ L_{∞} such that A_{U}u ∈ L_{∞}. Then, an application of Lemma 3.23(b) with 𝔥 = τ_{a} + U, 𝔥_{1} = τ + U, and A = A_{U} shows that τ_{p}(u) ⩾ −ω_{p}||u||_{2}^{2} for all 0 ⩽ u ∈ D(τ_{a} + U). This completes the proof since τ_{p} fulfils the first Beurling-Deny criterion.

So, let 0 ⩽ u ∈ D(A_{U}) ∩ L_{∞}, A_{U}u ∈ L_{∞}. Then u ∈ D(τ_{a} + U) ∩ L_{∞} and hence u^{r} ∈ D(τ_{a} + U) ∩ L_{∞}, ∇u^{r} = ru^{r−1}∇u for all r ⩾ 1. From this we easily obtain τ(u, u^{p−1}) = τ_{p}(u^{\frac{p}{2}} ) (cf. the computation on page 59) and thus, by the definition of A_{U}, ⟨A_{U}u, u^{p−1}⟩ = (τ_{p} + U)(u^{\frac{p}{2}} ). By Corollary 3.4(a) we have τ + U ↔ e^{−tA_{p,U}}. Since u, A_{U}u ∈ L_{2} ∩ L_{∞} ⊆ L_{p}, Lemma 3.24 implies that u ∈ D(A_{p,U}) and A_{p,U}u = A_{U}u. By (3.29) we obtain

(τ_{p} + U)(u^{\frac{p}{2}} ) = ⟨A_{p,U}u, u^{p−1}⟩ ⩾ ⟨(U − ω_{p})u^{p}⟩,

i.e., (3.30) holds.

Remark 3.25. Theorem 3.22 is in particular applicable in the case of weakly differentiable b_{1} and b_{2}. For j = 1, 2, we assume that b_{j} is of τ_{a}-regular divergence, i.e., there exists a measurable function div b_{j} such that |div b_{j}| is τ_{a}-regular and

2⟨b_{j}u, ∇u⟩ = −⟨(div b_{j})u^{2}⟩ (0 ⩽ u ∈ D(τ) ∩ Q(|div b_{j}|)).

Let U := V^{−} + |div b_{1}| + |div b_{2}|. Then

(τ_{1} + U)(u) = ⟨(−div b_{1} + V + U)u^{2}⟩ ⩾ 0,
(τ_{∞} + U)(u) = ⟨(div b_{2} + V + U)u^{2}⟩ ⩾ 0

for all 0 ⩽ u ∈ D(τ + U), so (T_{p})_{U} is L_{1}- and L_{∞}-contractive.

As an example, we are going to study the formal differential expression

−Δ + c_{1}|x|^{α}x · ∇ + c_{2}|x|^{α} on Ω := ℝ^{N},

with α, c_{1}, c_{2} ∈ ℝ. In Remark 3.6(c) we already studied the case α = 0 in slightly greater generality, so we will assume α ≠ 0 here. The case α = −2 will be of particular interest (see also Example 3.31 below). Before we proceed to the example we collect some facts about form bounds of the potential r^{α} with respect to −Δ. Here and in the following, r: ℝ^{N} → ℝ, r(x) := |x|.

Remark 3.26. Let a be the identity matrix, D(τ_{a}) = W_{2}^{1}(ℝ^{N}). Then τ_{a} is the standard Dirichlet form, τ_{a}(u) = ||∇u||_{2}^{2}. We will make use of the following Hardy inequality (see [KaWa72; Lemma 1]). If N ⩾ 3 then

||\frac{u}{r}||_{2} ⩽ \frac{2}{N − 2}||∇u||_{2} (u ∈ W_{2}^{1}(ℝ^{N})).

This inequality is sharp in the sense that there exist no β < \frac{2}{N−2}, c ∈ ℝ with ||\frac{u}{r}||_{2} ⩽ β||∇u||_{2} + c||u||_{2} for all u ∈ W_{2}^{1}(ℝ^{N}). Moreover, for N = 1, 2 there exist no β, c at all for which the inequality is valid. Expressed in the form language, Hardy’s inequality states that r^{−2} is form bounded with respect to τ_{a}, \frac{(N−2)^{2}}{4r^{2}} ⩽ τ_{a}.

Let now α < −2. Then for all ε > 0 there exists c_{ε} > 0 such that r^{−2} ⩽ εr^{α} + c_{ε}. Since Hardy’s inequality is sharp this implies that r^{α} is not form bounded with respect to τ_{a}, i.e., r^{α} ⩽̸ c(τ_{a} + 1) for all c > 0. The same holds for α > 0 since then |x|^{α} → ∞ as |x| → ∞.

Finally, let α ∈ (−2, 0). Then for all ε > 0 there exists c_{ε} > 0 such that r^{α} ⩽ εr^{−2} + c_{ε}. If N ⩾ 3 then Hardy’s inequality implies that r^{α} has zero form bound with respect to τ_{a}. The same holds for N = 2. This is an easy consequence of the Sobolev imbedding ||u||_{q} ⩽ c_{q}||u||_{2,1} for all q ∈ [2, ∞), u ∈ W_{2}^{1}(ℝ^{2}):

⟨r^{α}|u|^{2}⟩ ⩽ n||u||_{2}^{2} + ||(r^{α} − n)^{+}||_{( \frac{q}{2} )′}|||u|^{2}||_{\frac{q}{2}} ⩽ n||u||_{2}^{2} + ||(r^{α} − n)^{+}||_{( \frac{q}{2} )′}c_{q}||u||_{2,1}^{2}.

It remains to note that ||(r^{α} − n)^{+}||_{( \frac{q}{2} )′} → 0 as n → ∞ if q is sufficiently large.

Example 3.27. Let N ⩾ 2 and, as above, τ_{a} the standard Dirichlet form on ℝ^{N}. Let b_{2} = 0 and define b_{1}, V by

b_{1}(x) := b(x) := c_{1}|x|^{α}x, V (x) := c_{2}|x|^{α},

for some α, c_{1}, c_{2} ∈ ℝ, α ≠ 0. Then W_{1} = |b|^{2} and |V | are τ_{a}-regular, i.e., (bV’) is fulfilled. Let

τ(u, v) := ⟨∇u, ∇v⟩ + ⟨∇u, bv⟩ + ⟨V u, v⟩

on D(τ) := D(τ_{a} + |b|^{2} + |V |). If c_{1} = 0 then D(τ) = D(τ_{a} + r^{α}) (unless c_{2} = 0). If c_{2} ≠ 0 then D(τ) = D(τ_{a} + r^{2(α+1)}) since Q(r^{α}) ⊇ D(τ_{a} + r^{2(α+1)}) (for α > 0 and for α ⩽ −2 this is trivial; for α ∈ (−2, 0) it follows from Remark 3.26).

Observe that b is of τ_{a}-regular divergence,

2⟨bu, ∇u⟩ = −c_{1}(N + α)⟨r^{α}u^{2}⟩ (0 ⩽ u ∈ D(τ)).

(For 0 ⩽ u ∈ C_{c}^{1}(ℝ^{N} \ {0}) apply partial integration; for general u the claim follows by density.) For p ∈ [1, ∞] we thus obtain

τ_{p}(u) = \frac{4}{pp′} ||∇u||_{2}^{2} + (c_{2} − \frac{1}{p}c_{1}(N + α))⟨r^{α}u^{2}⟩ (0 ⩽ u ∈ D(τ)).  (3.31)

Now let us investigate for which values of α, c_{1}, c_{2} and for which p ∈ [1, ∞) the form τ is associated with a positive quasi-contractive C_{0}-semigroup on L_{p}(ℝ^{N}).

First we consider the three cases α < −2, α > 0, and N = 2 = −α. Then, by Remark 3.26, the potential r^{α} in not form bounded with respect to τ_{a}. Thus, the above implies that τ_{p} is bounded below if and only if c_{2} − \frac{1}{p}c_{1}(N + α) ⩾ 0, and τ_{p} ⩾ 0 in this case. By Theorems 3.2(a) and 3.22 we obtain, for p ∈ (1, ∞), that τ is associated with a quasi-contractive C_{0}-semigroup on L_{p} if and only if c_{2} − \frac{1}{p}c_{1}(N + α) ⩾ 0. Let I be the set of all p ∈ (1, ∞) fulfilling this inequality.

The set I is of one of the types (1, ∞), (1, p_{0}], [p_{0}, ∞), ∅. If c_{2} ⩾ (c_{1}(N + α))^{+} then I = (1, ∞); if c_{2} ⩽ (c_{1}(N + α)) ∧ 0 then I = ∅. If c_{1}(N + α) < c_{2} < 0 then I = (1, \frac{c_{1}}{c_{2}} (N + α)]; if 0 < c_{2} < c_{1}(N + α) then I = [\frac{c_{1}}{c_{2}} (N + α), ∞). In particular we obtain: if α = −N, c_{2} ⩾ 0 then, for arbitrary c_{1} ∈ ℝ, the form τ is associated with a consistent family of contractive C_{0}-semigroups on L_{p}, p ⩾ 1.

Now we consider the case α ∈ [−2, 0), (N, α) ≠ (2, −2). Of course, τ is associated with a contractive C_{0}-semigroup on L_{p} in the cases discussed above, but we obtain more. If α ∈ (−2, 0) then, by Remark 3.26, for all ε > 0 there exists c_{ε} > 0 such that τ_{p} ⩾ (\frac{4}{pp′} − ε)τ_{a} − c_{ε}. Thus, τ is associated with a consistent family of quasi-contractive C_{0}-semigroups on L_{p}, p > 1.

Now assume α = −2 (and N ⩾ 3). Then, by (3.31) and Hardy’s inequality,

τ_{p} ⩾ (\frac{4}{pp′} \frac{(N − 2)^{2}}{4} − \frac{c_{1}}{p} (N − 2) + c_{2}) r^{−2}.

Since Hardy’s inequality is sharp we obtain that τ_{p} is bounded below if and only if \frac{(N−2)^{2}}{pp′} − c_{1} \frac{N−2}{p} + c_{2} ⩾ 0, and τ_{p} ⩾ 0 in this case. Thus, by Theorems 3.2(a) and 3.22, τ is associated with a quasi-contractive C_{0}-semigroup on L_{p}, for p ∈ (1, ∞), if and only if \frac{(N−2)^{2}}{pp′} − c_{1} \frac{N−2}{p} + c_{2} ⩾ 0. For c_{2} = 0 this condition simplifies to \frac{N−2}{p′} −c_{1} ⩾ 0. Thus, if c_{1} ⩾ N −2 then there is no quasi-contractive semigroup on any L_{p} associated with τ. If c_{1} ∈ (0, N − 2) (or c_{1} ⩽ 0) then τ is associated with a contractive C_{0}-semigroup on L_{p} for all p ⩾ \frac{N−2}{N−2−c_{1}} (or p ⩾ 1).

Finally we show that, for α = −2, N ⩾ 3, the form τ studied above constitutes an example in which the interval in the L_{p}-scale obtained in Theorem 3.8 cannot be extended. So let, from now on, N ⩾ 3, b(x) := c_{1}|x|^{−2}x, V (x) := c_{2}|x|^{−2}. As above let r(x) = |x|, and

τ(u, v) = ⟨∇u, ∇v⟩ + ⟨∇u, bv⟩ + ⟨V u, v⟩

on D(τ) = W_{2}^{1}(ℝ^{N}).

Proposition 3.28. Assume that τ is associated with a C_{0}-semigroup e^{−tA_{p}} on L_{p}, for some p ⩾ 1. Then

D(A_{p}) ⊇ D_{p} := \{u ∈ L_{p}; Δu, \frac{1}{r}|∇u|, \frac{1}{r^{2}} u ∈ L_{p}\},

and A_{p}u = (−Δ + b · ∇ + V)u for all u ∈ D_{p}.

Proof. Define the operator L in L_{p} by Lu := (−Δ + b · ∇ + V)u, D(L) := D_{p}. Let U := |b|^{2} + |V |. Then τ + U is a closed sectorial form in L_{2}. Let A_{U} be the associated m-sectorial operator in L_{2}. By Proposition 1.19, e^{−tA_{U}} extrapolates to a C_{0}-semigroup T_{U,p}(t) = e^{−tA_{U,p}} on L_{p}, and T_{p} = (T_{U,p})_{−U}.

Let first u ∈ C_{c}^{∞}(ℝ^{N} \ {0}). Then u ∈ D(τ + U) and

(τ + U)(u, v) = ⟨(L + U)u, v⟩ (v ∈ D(τ + U)).

Hence u ∈ D(A_{U}) and A_{U}u = (L + U)u. Since u, A_{U}u ∈ L_{p}, Lemma 3.24 implies that u ∈ D(A_{U,p}), A_{U,p}u = (L + U)u. Thus, A_{U,p} and L + U coincide on C_{c}^{∞}(ℝ^{N} \{0}). By [Voi86; Cor. 2.7] we have A_{p} ⊇ A_{U,p} −U, so A_{p} and L coincide on C_{c}^{∞}(ℝ^{N} \ {0}).

Let now u ∈ D_{p} ∩ L_{p,c}(ℝ^{N} \ {0}), ρ_{n} the standard mollifier. Then ρ_{n} ∗ u ∈ C_{c}^{∞}(ℝ^{N} \ {0}) for large n, and it is easy to see that ρ_{n} ∗ u → u, L(ρ_{n} ∗ u) → Lu in L_{p}. This implies A_{p}u = Lu since A_{p} is a closed operator.

Finally, let u ∈ D_{p}. Let φ ∈ C_{c}^{∞}(ℝ^{N}) such that 0 ⩽ φ ⩽ 1, φ↾_{B(0,1)} = 1, supp φ ⊆ B(0, 2). Let φ_{n}(x) := φ(\frac{x}{n}) · (1 − φ)(2nx) for x ∈ ℝ^{N}. Then φ_{n} ∈ C_{c}^{∞}(ℝ^{N} \ {0}), 0 ⩽ φ_{n} ⩽ 1, and φ_{n} → 1 a.e. as n → ∞. Let

B_{n} := (B(0, 2n) \ B(0, n)) ∪ (B(0, \frac{1}{n}) \ B(0, \frac{1}{2n})).

It is straightforward that

|∇φ_{n}| ⩽ \frac{2}{r}||∇φ||_{∞}χ_{B_{n}}, |Δφ_{n}| ⩽ \frac{4}{r^{2}} ||Δφ||_{∞}χ_{B_{n}} (n ∈ ℕ).

Thus, by dominated convergence we obtain that

L(φ_{n}u) = φ_{n}Lu − 2∇φ_{n} · ∇u + (b · ∇φ_{n} − Δφ_{n})u → Lu in L_{p}

as n → ∞. So u ∈ D(A_{p}) and A_{p}u = Lu since φ_{n}u → u in L_{p}.

Corollary 3.29. Assume that τ is associated with a C_{0}-semigroup e^{−tA_{p}} on L_{p}, for some p ∈ [1, ∞). Let σ < \frac{N}{p} − 2, u := r^{−σ}e^{− \frac{r^{2}}{2}} . Then u ∈ D(A_{p}) and

A_{p}u = (−\frac{1}{r^{2}} (σ^{2} − (N − 2 − c_{1})σ − c_{2}) + N − c_{1} − 2σ − r^{2})u.

Proof. By Proposition 3.28 we have u ∈ D(A_{p}) and A_{p}u = (−Δ + b · ∇ + V)u since u ∈ D_{p}. The second assertion now results from a direct computation.

The next extrapolation lemma is a modification of the result from [Cou91] with literally the same proof.

Lemma 3.30. Let p_{0} ⩽ p < q ⩽ p_{1}. Let T be a semigroup on L_{p} satisfying ||T (t)||_{p_{0}→p_{0}} ⩽ C, ||T (t)||_{p_{1}→p_{1}} ⩽ C, and

||T (t)||_{p→q} ⩽ Ct^{−α( \frac{1}{p} − \frac{1}{q})} (t > 0),

for some C ⩾ 1, α > 0. Then

||T (t)||_{p_{0}→p_{1}} ⩽ Ct^{−α( \frac{1}{p_{1}} − \frac{1}{p_{0}} )} (t > 0).

Example 3.31. Let c_{1} := β \frac{N−2}{2} and c_{2} := −γ \frac{(N−2)^{2}}{4} with 0 ⩽ β < 2, 0 < γ < (1 − \frac{β}{2} )^{2}. By (3.31) we have

τ_{p}(u) = \frac{4}{pp′} ||∇u||_{2}^{2} − \frac{(N−2)^{2}}{4} (\frac{2}{p}β + γ)||\frac{u}{r}||_{2}^{2} (u ∈ D(τ)).

Hence, by Hardy’s inequality,

τ_{p}(u) ⩾ \frac{(N−2)^{2}}{4} (4\frac{1}{p}(1 − \frac{1}{p}) − 2β \frac{1}{p} − γ)||\frac{u}{r}||_{2}^{2} (u ∈ D(τ)).

Let 1 < p_{−} < p_{+} < ∞ such that \frac{1}{p_{±}} are the roots of the equation

4x(1 − x) − 2βx − γ = −4(x^{2} − (1 − \frac{β}{2} )x + \frac{γ}{4}) = 0.  (3.32)

Then τ_{p} is bounded below if and only if p ∈ [p_{−}, p_{+}]. Hence, by Theorem 3.8, τ is associated with a consistent family of C_{0}-semigroups e^{−tA_{p}} on L_{p},

p_{min} := (\frac{N}{N−2}p′_{−})′ < p < \frac{N}{N−2}p_{+} =: p_{max}.

By Theorems 3.2(a) and 3.22, e^{−tA_{p}} is quasi-contractive if and only if p ∈ [p_{−}, p_{+}].

We are going to show that, for q ∉ (p_{min}, p_{max}), the form τ is not associated with a C_{0}-semigroup on L_{q}. Let

σ := \frac{N}{p_{max}} = \frac{N − 2}{p_{+}}, p_{0} := \frac{N}{σ + 2} = (\frac{N}{N−2}p′_{+})′.

Then \frac{σ}{N−2} is a root of equation (3.32). Hence, σ^{2} − (N − 2 − β \frac{N−2}{2})σ + γ \frac{(N−2)^{2}}{4} = 0. Observe that p_{0} ∈ (p_{min}, p_{+}). Let u := r^{−σ}e^{− \frac{r^{2}}{2}} , c := N − β \frac{N−2}{2} − 2σ, p ∈ (p_{min}, p_{0}). Then σ < \frac{N}{p} − 2. By Corollary 3.29, A_{p}u = (c − r^{2})u ⩽ cu and hence u ⩽ e^{t(c−A_{p})}u for all t ⩾ 0.

Now assume that τ is associated with a C_{0}-semigroup T_{q} on L_{q}, for some q ⩾ p_{max}. Then T_{p}, T_{q} are consistent by Proposition 1.22(a). By (3.24) and Lemma 3.30, e^{−tA_{p}}: L_{p} → L_{q} for all t > 0. In particular, e^{−tA_{p}}u ∈ L_{q}. Since e^{t(c−A_{p})}u ⩾ u, this contradicts the fact that u ∉ L_{q} (recall σ = \frac{N}{p_{max}} ⩾ \frac{N}{q}). Considering the adjoint semigroup we show that e^{−tA_{p}} does not extrapolate to a semigroup on L_{q}, for any q ⩽ p_{min}.

In the case of Schrödinger semigroups, i.e. β = 0, this example was first given in [KPS81]. More precisely, for a certain class of potentials it was shown that the Schrödinger semigroup acts on L_{p}(ℝ^{N}) for p ∈ (p_{min}, p_{max}), and it was claimed that for potentials of the type \frac{c}{r^{2}} with c < 0 the interval is maximal. A strict proof of this claim was given by Yu. Semenov (private communication).

Finally, let us return to the remark in the paragraph following the proof of Proposition 1.15. There the following was claimed. Given a densely defined closed sectorial form τ fulfilling the first Beurling-Deny criterion, T the associated C_{0}- semigroup, and V ⩾ 0 measurable, the T -admissibility of −V does not imply V ⩽ Re τ + ω for some ω ∈ ℝ. We now show that the above constitutes an example.

Let 0 < β < 1. Then there exists 0 < γ < (1 − \frac{β}{2} )^{2} such that p_{min} < 2 < p_{−}, where p_{min}, p_{−} are defined as above. We have shown that τ is associated with a positive C_{0}-semigroup T on L_{2} which is not quasi-contractive. Define the form τ_{0} in L_{2} by

τ_{0}(u, v) := ⟨∇u, ∇v⟩ + ⟨∇u, bv⟩

on D(τ_{0}) = W_{2}^{1}(ℝ^{N}). Then τ_{0} is a densely defined closed sectorial form fulfilling the first Beurling-Deny criterion. Let T_{0} be the associated (contractive) C_{0}- semigroup on L_{2}. Let U := −V = γ \frac{(N−2)^{2}}{4r^{2}} . Then τ_{0} = τ + U. Since τ is associated with the C_{0}-semigroup T on L_{2}, Proposition 1.19 implies that −U is T_{0}-admissible. But we do not have U ⩽ Re τ_{0} + ω for some ω ∈ ℝ: this would imply τ_{2} = Re τ = Re τ_{0} − U ⩾ −ω, contradicting 2 ∉ [p_{−}, p_{+}].

We point out that, in the above situation, τ is a form that is not sectorial—it is not even bounded from the left. Nevertheless, by means of Definition 1.20, τ is associated with a C_{0}-semigroup on L_{2}.

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Affirmation

Hereby I affirm that I wrote the present thesis without any inadmissible help by a third party and without using any other means than indicated. Thoughts that were taken over directly or indirectly from other sources are indicated as such. This thesis has not been presented to any other examination board in this or a similar form, neither in this or any other country.

The present thesis was written at Dresden University of Technology, Institute of Analysis, under the supervision of Prof. Dr. Jürgen Voigt.

Versicherung

Hiermit versichere ich, daß ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht. Die Arbeit wurde bisher weder im Inland noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde vorgelegt.

Die vorliegende Arbeit wurde an der Technischen Universität Dresden, am Institut für Analysis, unter der Betreuung von Herrn Prof. Dr. Jürgen Voigt angefertigt.

Dresden, 15. 2. 2001