On the L_{p}-theory of C_{0}-semigroups associated with second order elliptic operators. I

Zeev Sobol

School of Mathematics University of Bristol Bristol BS8 1TW United Kingdom

Hendrik Vogt

Institut für Analysis Fachrichtung Mathematik Technische Universität Dresden Dresden, Germany

Abstract

We study L_{p}-theory of second order elliptic divergence type operators with measurable coefficients. To this end, we introduce a new method of constructing positive C_{0}-semigroups on L_{p} associated with sesquilinear (not necessarily sectorial) forms in L_{2}. A precise condition ensuring that the elliptic operator is associated with a quasi-contractive C_{0}-semigroup on L_{p} is established.

1 Introduction and main results

In this paper we study 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. The aim of the paper is to construct a quasi-contractive 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 is well-known, this implies well-posedness of the corresponding Cauchy problem.

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 is

τ(u, v) := ⟨a∇u, ∇v⟩ + ⟨∇u, b_{1}v⟩ − ⟨b_{2}u, ∇v⟩ + ⟨V u, v⟩  (1.1)

on a suitable domain D(τ) corresponding to the boundary conditions. (Here and in the sequel, ⟨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.)

The traditional way of constructing the corresponding C_{0}-semigroup is the following. If the form τ is densely defined, sectorial and closed then it is associated with an m-sectorial operator A in L_{2} which generates a quasi-contractive analytic semigroup e^{−At} on L_{2} (cf. [4, Thm. VI.2.1]). If ||e^{−At}↾_{L_{2}∩L_{p}} ||_{L_{p}→L_{p}} ⩽ Me^{ω_{p}t} for some p ∈ [1, ∞), then the semigroup extends to a semigroup T_{p} on L_{p}. In this case we say that e^{−At} extrapolates to the semigroup T_{p} on L_{p}, which is consistent with e^{−At} in the sense that e^{−At}↾_{L_{2}∩L_{p}} = T_{p}(t)↾_{L_{2}∩L_{p}} for all t ⩾ 0. For p > 1, the semigroup T_{p} is always strongly continuous, whereas for p = 1 this is the case if, e.g., T_{1} is positive or quasi-contractive (see [19]). The above approach was used for constructing semigroups acting in all L_{p}, 1 ⩽ p < ∞ (this case is welldocumented, see, e.g., [14] and [3]), as well as for constructing semigroups acting in L_{p} only for p from some subinterval of [1, ∞) containing 2; see, e.g., [2], [6].

However, we do not assume τ to be a sectorial form in L_{2}; even its real part need not be bounded below, so that the traditional approach is not applicable. In the case b_{2} = 0 and V = 0, non-sectorial forms have been studied in [5], [6] where the coefficients of the first order terms of L are approximated in such a way that the approximating forms become sectorial in L_{2} and the corresponding semigroups converge to a C_{0}-semigroup, in a suitable L_{p}.

In this paper we develop a new approach to the construction of a quasicontractive C_{0}-semigroup associated with the form τ, which even in L_{2} gives rise to a C_{0}-semigroup 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.

Instead, 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 τ, it makes the sum sectorial in L_{2}. The sequence of the approximating semigroups T_{m}, which are associated with the sectorial forms τ_{m} := τ +U −U ∧m (m ∈ ℕ), extrapolates to a suitable L_{p} and strongly converges to a quasi-contractive C_{0}-semigroup on L_{p}. The use of the perturbation theory of positive semigroups developed in [17], [18] is crucial for the realization of this idea.

The approach we present is in fact a general method of constructing positive C_{0}-semigroups on L_{p} corresponding to sesquilinear forms in L_{2} (see Section 3 for details). In the context of Schrödinger operators with magnetic fields, and dominated semigroups with singular complex potentials, a similar approximation idea was used in [12] and in [7].

The result we obtain is sharp in the sense that, for a wide class of coefficients, the sufficient condition (see estimate (1.3) below) for the validity of our main theorem becomes necessary (see Section 6 for details).

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

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

|Im ζ^{∗}aζ| ⩽ α Re ζ^{∗}aζ a.e. (ζ ∈ ℂ^{N})

for some α ⩾ 0 (where ζ^{∗} is the transpose of \overline{ζ}). Let a_{s} := \frac{a+a^{⊤}}{2}. Then

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

defines a closed sectorial (non-symmetric) Dirichlet form in L_{2} (for the closedness cf. [13, Theorem 3.2]). 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, i.e., D(τ_{a}) ∩ Q(W_{j}) and D(τ_{a}) ∩ Q(|V |) are cores for τ_{a}. (For a potential U ⩾ 0, Q(U) := {u ∈ L_{2}; U|u|^{2} ∈ L_{1}} denotes the domain of the form U(u) = ⟨U|u|^{2}⟩ in L_{2}.)

We define the form τ on D(τ) := D(τ_{a}) ∩ Q(W_{1} + W_{2} + |V |) by (1.1). This is possible since for 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}.  (1.2)

Furthermore, D(τ) is dense in D(τ_{a}) as can be seen from Lemma 3.13 below. In particular, τ is densely defined.

Although the form τ itself need not be sectorial, the form τ + U with domain D(τ) ∩ Q(U) is sectorial and closed for all U ⩾ U_{0} := W_{1} + W_{2} + 2V^{−} since the sum of the first order terms of τ is form small with respect to τ_{a} + W_{1} + W_{2} by (1.2).

The only quantitative condition we need is obtained from the Lumer-Phillips theorem by a formal computation. Suppose τ is associated with a positive quasicontractive C_{0}-semigroup T_{p}(t) = e^{−A_{p}t} on L_{p}, for some p ∈ [1, ∞). 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} in case p > 1, and to ⟨A_{1}u⟩ ⩾ −ω_{1}||u||_{1} in case p = 1, for some ω_{p} ∈ ℝ and all 0 ⩽ u ∈ D(A_{p}). Formally, A_{p}u = Lu, ∇u^{p−1} = \frac{2}{p′} u^{p/2−1}∇u^{p/2}, and ∇u = \frac{2}{p}u^{1−p/2}∇u^{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^{p/2}, ∇u^{p/2}⟩ + ⟨(\frac{2}{p}b_{1} − \frac{2}{p′} b_{2})u^{p/2}, ∇u^{p/2}⟩ + ⟨V u^{p}⟩

in case p > 1 and, in case p = 1,

⟨A_{1}u⟩ = ⟨−∇ · (a∇u) + b_{1} · ∇u + ∇ · (b_{2}u) + V u⟩
= 2⟨∇u^{1/2}, b_{1}u^{1/2}⟩ + ⟨V u⟩.

Now we define quadratic forms τ_{p} on D(τ_{p}) := D(τ) (1 ⩽ p < ∞),

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

Then the natural condition for L_{p}-accretivity is

τ_{p}(u) ⩾ −ω_{p}||u||_{2}^{2} (u ∈ D(τ)),  (1.3)

i.e., τ_{p} is bounded from below. Note that τ_{2} = Re τ (as to be expected), where the form Re τ is defined by (Re τ)(u, v) := \frac{1}{2}(τ(u, v) + \overline{τ(v, u)}) on D(Re τ) := D(τ).

The construction of the C_{0}-semigroup on L_{p}, corresponding to the formal differential expression L with boundary conditions prescribed by D(τ_{a}), is given in the following theorem, which constitutes a simplified version of the main result of the paper, Theorem 4.2.

Theorem 1.1. Let assumptions (a) and (bV) be fulfilled. Let U_{0} := W_{1} + W_{2} + 2V^{−}, and let T_{0,2} be the C_{0}-semigroup on L_{2} associated with the form τ + U_{0}. Let I be the set of all p ∈ [1, ∞) such that ω_{p} := inf{ω ∈ ℝ; τ_{p} ⩾ −ω} < ∞. Then the following assertions hold.

(i) The set I is an interval in [1, ∞), and T_{0,2} extrapolates to a C_{0}-semigroup T_{0,p}(t) = e^{−A_{0,p}t} on L_{p}, for all p ∈ I.

(ii) For all p ∈ I, the sequence of C_{0}-semigroups T_{m,p}(t) = e^{−(A_{0,p}−U_{0}∧m)t} strongly converges in L_{p} to a C_{0}-semigroup T_{p}(t) = e^{−A_{p}t} satisfying ||T_{p}(t)|| ⩽ e^{ω_{p}t}. For p, q ∈ I, the semigroups T_{p} and T_{q} are consistent.

(iii) For all p ∈ I \ {1}, the form τ_{p} is closable. For all u ∈ D(A_{p}) we have |u|^{p/2} sgn u ∈ D(\overline{τ_{p}}) and

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

(iv) If, in addition, we assume that

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

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

We shall call A_{p} the m-accretive operator in L_{p}, T_{p} the quasi-contractive C_{0}- semigroup on L_{p} associated with the form τ. The operator A_{p} is an L_{p}-realization of L with boundary conditions prescribed by D(τ_{a}).

Remarks 1.2. (a) In fact, as it will be shown in the main body of the paper (see Corollary 4.4 below), the semigroups T_{p} constructed in the theorem do not depend on the approximating sequence of potentials. Furthermore, the assertions hold with U_{0} replaced by any positive τ_{a}-regular potential U such that τ + U is sectorial and closable in L_{2}.

(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)}} (note that τ̃_{a} is a Dirichlet form). Then assumptions (a) and (bV) are fulfilled with τ̃_{a} in place of τ_{a}, so Theorem 4.2 is still applicable to the form τ.

(c) If the form τ itself is sectorial then it is closable (see Lemma 3.5 below). In this case we have 2 ∈ I, A_{2} is the m-sectorial operator associated with \overline{τ} and, for f ∈ L_{2}, the function u(t) := T_{2}(t)f is the weak solution of the Cauchy problem

\begin{cases}u_{t} & = −Lu, \\ u(0) & = f \end{cases}

with boundary conditions prescribed by D(τ).

(d) Let us point out that the interval I given in Theorem 4.2 is a set of p ∈ [1, ∞) for which the form τ is associated with a quasi-contractive C_{0}-semigroup T_{p} on L_{p} (I \ {1} is the maximal set of such p ∈ (1, ∞) under the conditions of Corollary 6.4 below). The set of all p ∈ [1, ∞) such that τ is associated with a C_{0}-semigroup T_{p} on L_{p} can be strictly larger than I, see [8].

The remainder of the paper is organized as follows. In Section 2 we give a brief account of Voigt’s perturbation theory for positive semigroups. In Section 3 we show how to associate a positive C_{0}-semigroup on L_{p}(μ) with a sesquilinear form in L_{2}(μ). Section 4 contains the precise formulation of the main theorem and some useful consequences of it. The proof of the main theorem is given in Section 5. In Section 6 we discuss the sharpness of the main result.

2 Perturbations of positive C_{0}-semigroups by real-valued potentials

In this section we give a short survey of J. Voigt’s perturbation theory for positive C_{0}-semigroups developed in [17], [18].

Let (Ω, μ) be a measure space, 1 ⩽ p < ∞. Let T be a positive C_{0}-semigroup on L_{p}(μ), i.e., the semigroup operators T (t) (t ⩾ 0) are positivity preserving. Let −A be the generator of T and V: Ω → ℝ a measurable function. If V ∈ L_{∞}(μ) then T_{V} denotes the C_{0}-semigroup generated by −(A + V).

The definition of T_{V} is extended to unbounded real-valued potentials by approximating V by V^{(n)} := (V ∧ n) ∨ (−n) and letting

T_{V} (t) := s-lim_{n→∞} T_{V^{(n)}}(t) (t ⩾ 0)  (2.1)

if the limits exist. Obviously, T_{V} is a semigroup in this case. If V ⩾ 0 then (T_{V^{(n)}}) is a monotone decreasing sequence, for V ⩽ 0 it is monotone increasing. This leads to the following definition.

Definition 2.1. ([17, Def. 2.2], [18, Def. 2.5], [18, Def. 3.1])

(a) If V ⩾ 0 then the limit in (2.1) exists for all t ⩾ 0. If T_{V} is strongly continuous, V is called T -admissible. In this case, T_{V^{(n)}} → T_{V} as n → ∞, i.e., T (t)f = lim_{n→∞} T_{V^{(n)}}(t)f, uniformly for t in bounded subsets of [0, ∞), for all f ∈ L_{p}.

(b) If V ⩽ 0 then V is called T -admissible if the limit in (2.1) exists for all t ⩾ 0 and defines a C_{0}-semigroup. In this case, T_{V^{(n)}} → T_{V} as n → ∞.

By [18, 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. If T = (T_{V})_{−V}, then V is called T -regular.

The following result expresses, roughly speaking, that negative admissible potentials are always regular.

Lemma 2.2. (cf. [18, Thm. 2.6, Prop. 3.3(b)]) Let V ⩾ 0 be measurable. If −V is T -admissible, then (T_{−V})_{V} = T, and V is T -regular.

Lemma 2.3. ([17, Prop. 3.1]) Let p, q ∈ [1, ∞), T_{p}, T_{q} consistent positive C_{0}- semigroups on L_{p}(μ), L_{q}(μ), respectively, V ⩾ 0 measurable.

(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.

We conclude the section with the following approximation result which we will use in Section 4 to show that the semigroup constructed in [6, Thm. 6] coincides with the semigroup constructed in Theorem 1.1.

Proposition 2.4. Let p ∈ (1, ∞). Let T_{n} (n ∈ ℕ ∪ {∞}) be positive C_{0}-semi-groups on L_{p}(μ) with T_{n} → T_{∞}. Let 0 ⩽ V ∈ (L_{1} + L_{∞})(μ) such that −V is T_{n}-admissible (n ∈ ℕ ∪ {∞}), and

||(T_{n})_{−V} (t)||_{p→p} ⩽ e^{ωt}, ||(T_{n})_{−V} (t)||_{∞→∞} ⩽ Ce^{ωt} (n ∈ ℕ, t ⩾ 0).

for some ω ∈ ℝ, C ⩾ 1. Then (T_{n})_{−V} → (T_{∞})_{−V}.

The crucial idea of the proof is to make use of the following result which gives an explicit rate of the convergence T_{−V ∧n} → T_{−V}.

Lemma 2.5. Let p ∈ (1, ∞), T be a positive C_{0}-semigroup on L_{p}(μ), and 0 ⩽ V ∈ (L_{1} +L_{∞})(μ) such that −V is T -admissible, and T_{−V} is contractive in L_{p}(μ) and bounded in L_{∞}(μ). Let −A be the generator of T, −A_{−V} the generator of T_{−V}. Then

||(λ + A_{−V})^{−1}f − (λ + A − V ∧ n)^{−1}f||_{p} ⩽ Cλ^{−1−1/p}||(V − n)^{+}||_{1}^{1/p}||f||_{∞}

for all 0 ⩽ f ∈ (L_{p} ∩ L_{∞})(μ), λ > 0 and n ∈ ℕ such that (V − n)^{+} ∈ L_{1}(μ), where C is the L_{∞}-bound of T_{−V}.

Proof. Let f, λ, n be given. For m ∈ ℕ let V_{m} := V ∧ m. Then

u_{m} := (λ + A − V_{m})^{−1}f ↑ u := (λ + A_{−V})^{−1}f as m → ∞,

and ||u||_{∞} ⩽ \frac{C}{λ} ||f||_{∞}. For m ∈ ℕ we have

(λ + A − V_{m})^{−1} − (λ + A − V_{n})^{−1} = (λ + A − V_{m})^{−1}(V_{m} − V_{n})(λ + A − V_{n})^{−1}

and therefore (λ + A − V_{m})(u_{m} − u_{n}) = (V_{m} − V_{n})u_{n}. The contractivity of T_{−V} implies that A − V_{m} is accretive, so we obtain, for m ⩾ n,

λ||u_{m} − u_{n}||_{p}^{p} ⩽ ⟨(λ + A − V_{m})(u_{m} − u_{n}), (u_{m} − u_{n})^{p−1}⟩
= ⟨(V_{m} − V_{n})u_{n}, (u_{m} − u_{n})^{p−1}⟩
⩽ ⟨(V − V_{n})u^{p}⟩ ⩽ ||(V − n)^{+}||_{1}||u||_{∞}^{p}.

We conclude that ||u_{m} − u_{n}||_{p}^{p} ⩽ λ^{−p−1}||(V − n)^{+}||_{1}(C||f||_{∞})^{p}, and m → ∞ completes the proof.

Proof of Proposition 2.4. Without restriction assume ω = 0. Let −A_{n}, −(A_{n})_{−V} be the generators of T_{n}, (T_{n})_{−V}, respectively. By the assumption, A_{n} → A_{∞} in the strong resolvent sense as n → ∞. So A_{n} −V ∧m → A_{∞} −V ∧m in the strong resolvent sense as n → ∞, for all m ∈ ℕ. By Lemma 2.5 we know that A_{n} − V ∧ m → (A_{n})_{−V} in the strong resolvent sense as m → ∞, uniformly in n ∈ ℕ. Since A_{∞} − V ∧ m → (A_{∞})_{−V} in the strong resolvent sense, this yields the desired conclusion.

3 The first Beurling-Deny criterion for sesquilinear forms

It is well-known that with every densely defined closed sectorial form in a Hilbert space H one can associate an analytic semigroup on H. In this section we are going to present a procedure how 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 below.

Definition 3.1. 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 fulfill the first Beurling-Deny criterion if τ is real and u^{+} ∈ D(τ), τ(u^{+}, u^{−}) ⩽ 0 for all real-valued u ∈ D(τ).

Note that, if τ fulfills the first Beurling-Deny criterion then so does Re τ. The following proposition, due to Ouhabaz ([11, Prop. 2.2 and Thm. 2.4]),

shows the relevance of these two notions.

Proposition 3.2. 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 reality preserving) if and only if τ is real, and T is positive if and only if τ fulfills the first Beurling-Deny criterion.

The next lemma states that it suffices to verify the conditions of Definition 3.1 on a form core.

Lemma 3.3. Let τ be a closable sectorial form. If τ fulfills 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 [9, 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 characterizes admissibility of potentials via form conditions, in the case of symmetric forms.

Proposition 3.4. (cf. [17, 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 T_{V} is associated with τ + 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 T_{−V} is associated with \overline{τ − V}.

Proof. All the assertions of the proposition, except for the closability of τ − V, are shown in [17]. 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 ([10, 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, (T_{−V})_{V} = T is associated with both τ̃ + V and τ, taking into account Lemma 2.2 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 3.4(a) is valid even for sectorial forms, see [10, Kor. 12.1.4(a)]. It is clear that a sesquilinear form τ fulfills 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 3.4(b).

Corollary 3.5. (cf. [10, 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 3.4(b) implies that \overline{τ + V} − V is closable. Thus, τ is closable since τ ⊆ \overline{τ + V} − V.

Definition 3.6. 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).

Remark 3.7. (a) For example, V ∈ (L_{1} + L_{∞})(μ) is τ-regular if τ is a Dirichlet form, since D(τ) ∩ L_{∞}(μ) ⊆ Q(V) is a core for τ.

(b) 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 [15]).

The following lemma states in particular that form regularity implies semigroup regularity.

Lemma 3.8. 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} is associated with \overline{τ + V}.

Proof. Note that, by Lemma 3.3, \overline{τ + V} fulfills the first Beurling-Deny criterion. Let T_{1} be the positive C_{0}-semigroups 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 [4, Thm. VIII.3.6] to obtain (T_{1})_{−V ∧n} → T. Thus, −V is T_{1}-admissible, and (T_{1})_{−V} = T. Lemma 2.2 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 [18, Prop. 3.4(a)].

In [10, 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. It is fundamental for Section 4.

Proposition 3.9. Let U ⩾ 0 be measurable, Q(U) ⊇ D(τ), τ + U sectorial and closable, T_{U,2} the positive C_{0}-semigroup associated with \overline{τ + U}. Let V ⩾ 0 be (τ + U)-regular, τ + V sectorial and closable, T_{V,2} the positive C_{0}-semigroup associated with \overline{τ + V}. 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 [18, Prop. 3.4(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 3.8 implies that both (T_{V,2})_{U} and (T_{U,2})_{V} are associated with \overline{(τ + V) + U} = \overline{(τ + U) + 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 2.3(c). Since −U is T_{U,p}-admissible we obtain by [18, Thm. 2.6] 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 2.3(a) and (b) that T_{V,2} and (T_{p})_{V} are consistent.

Proposition 3.9 leads to the following definition. Recall that τ is a densely defined sesquilinear form fulfilling the first Beurling-Deny criterion.

Definition 3.10. 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 with Q(U) ⊇ D(τ) such that 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 3.9, the semigroup T_{p} is uniquely determined by the form τ. If τ itself is sectorial and closable, we can choose U = 0. In this case T_{2}(t) = e^{−At} where A is the m-sectorial operator associated with τ by the first representation theorem (see [4, Thm. VI.2.1]).

The following result is a generalization of Lemma 3.8.

Proposition 3.11. Let p ∈ [1, ∞) and assume that τ is associated with a positive C_{0}-semigroup T_{p} on L_{p}(μ). Let U ⩾ 0 with Q(U) ⊇ D(τ) be such that 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 3.9 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 2.2 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 [18, Prop. 3.3(a)]. Moreover, −U is admissible with respect to (T_{p})_{U+V} and ((T_{p})_{U+V} )_{−U} = (T_{p})_{V}, by [18, Thm. 3.4]. Hence τ +V ↔ (T_{p})_{V}.

Given τ, we consider the adjoint form τ^{∗} which is defined by

τ^{∗}(u, v) := \overline{τ(v, u)} on D(τ^{∗}) := D(τ).

Proposition 3.12. Let p ∈ (1, ∞) and assume that τ is associated with a positive C_{0}-semigroup T_{p} on L_{p}(μ). Then the form τ^{∗} is associated with the adjoint semigroup T_{p}^{∗} on L_{p′}(μ).

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 3.12. Let U ⩾ 0 with Q(U) ⊇ D(τ) such that 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}.

It is easy to see that τ^{∗} + U is closable, fulfills 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}^{∗}.

We conclude the section by a result needed for applications of Proposition 3.9.

Lemma 3.13. Let τ be sectorial and closable, U, V ⩾ 0 measurable. Assume that U is τ-regular. Then V is τ-regular if and only if V is (τ + U)-regular. As a consequence, U + V is τ-regular if U, V are τ-regular.

Proof. Let V be (τ + U)-regular. Then D((τ + U) + V) is a core for τ + U and hence a core for τ. Therefore, D(τ + V) is a core for τ, i.e., V is τ-regular.

Conversely, assume that V is τ-regular. Without restriction, τ is symmetric and τ ⩾ 0. Let 0 ⩽ u ∈ D(τ + U). There exists (u_{n}) ⊆ D(τ + V) such that u_{n} → u in D(τ) as n → ∞. Let v_{n} := (Re u_{n})^{+}. Since τ fulfills the first Beurling-Deny criterion we have lim sup_{n→∞} τ(v_{n}) ⩽ lim_{n→∞} τ(u_{n}) = τ(u). The lower semicontinuity of τ implies that v_{n} → u in D(τ) as n → ∞. Moreover, τ((u − v_{n})^{+}) ⩽ τ(u − v_{n}) → 0 and thus u ∧ v_{n} = u − (u − v_{n})^{+} → u in D(τ) as n → ∞. Finally, u ∧ v_{n} → u in Q(U) by Lebesgue’s dominated convergence theorem. We infer that D((τ + U) + V) ∋ u ∧ v_{n} → u in D(τ + U). This shows that D((τ + U) + V) is a core for τ + U.

4 L_{p}-properties of elliptic differential operators

In this section we formulate the main result of the paper and deduce some corollaries. We refer to Section 1 for the notation.

Recall that the form τ is defined on D(τ) := D(τ_{a})∩Q(W_{1}+W_{2}+|V |) by (1.1). Since τ_{a} is a Dirichlet form, (Re u)^{+} ∈ D(τ) for all u ∈ D(τ). Therefore, τ fulfills the first Beurling-Deny criterion (we actually have τ(u^{+}, u^{−}) = 0 for all realvalued u ∈ D(τ), and τ(u, v) ∈ ℝ for all real-valued u, v ∈ D(τ).) Further, D(τ) is a core for τ_{a} by Lemma 3.13, in particular, τ is densely defined.

The forms τ_{p} play a crucial role in all our results on elliptic operators. We will also make use of the symmetric form τ_{∞} defined by

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

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 results.

Proposition 4.1. 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} + 2V^{−}, the form τ + U is sectorial and closed. For all 1 < p < ∞ and U ⩾ p′W_{1} + pW_{2} + 2V^{−}, the symmetric form τ_{p} + U is non-negative and 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, for some 1 ⩽ p_{0} < p < p_{1} ⩽ ∞, we have τ_{p_{j}} ⩾ −ω_{p_{j}} (j = 0, 1) then we can choose ε_{p} = 4(\frac{1}{p_{0}} − \frac{1}{p})(\frac{1}{p} − \frac{1}{p_{1}} ), c_{p} = θω_{p_{0}} + (1 − θ)ω_{p_{1}}, with

θ = \frac{p_{0}^{−1}−p^{−1}}{p_{0}^{−1}−p_{1}^{−1}} .

(c) For all p, q ∈ ^{∘} I, the norms on the Hilbert spaces D(\overline{τ_{p}}) and D(\overline{τ_{q}}) are equivalent.

Proof. (a) From (1.2) 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} + 2V^{−}. The same argument works for τ_{p} if 1 < p < ∞. By Corollary 3.5 we obtain that τ_{p} is closable if it is bounded below.

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}.  (4.1)

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 (4.1) 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 4.2 includes cases where τ is not even bounded from the left. However, the form τ +W_{1} +W_{2} +2V^{−} is sectorial and closed by Proposition 4.1(a). This enables us to make use of Definition 3.10 in the main result of the paper which reads as follows.

Theorem 4.2. 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 τ is associated with a consistent family of positive C_{0}-semigroups T_{p} on L_{p} with ||T_{p}(t)|| ⩽ e^{ω_{p}t} for all p ∈ I, t ⩾ 0.

Let −A_{p} be the generator of T_{p} (p ∈ I). Then, for all p ∈ I \ {1} and u ∈ D(A_{p}) we have v_{p} := u|u|^{p/2−1} = |u|^{p/2} sgn u ∈ D(\overline{τ_{p}}) and

Re⟨A_{p}u, u|u|^{p−2}⟩ ⩾ \overline{τ_{p}}(v_{p}).  (4.2)

If, in addition,

|Im⟨(b_{1} + b_{2})u, ∇u⟩| ⩽ c_{1}τ_{p}(u) + c_{2}||u||_{2}^{2} (u ∈ D(τ))  (4.3)

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}.

The proof of the theorem is delegated to Section 5.

Remarks 4.3. (a) 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.

(b) 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 quasi-contractive semigroup T_{∞} on L_{∞} with the form τ. Observe that the condition on τ_{∞} imposes no additional restriction on b_{1}.

(c) Lemma 4.1(b) demonstrates the relevance of inequality (4.2): Assume that the domain of τ_{a} admits Sobolev imbedding, i.e., D(τ_{a}) ⊆ L_{2j} for some j > 1. Then it is easy to show that, for all p ∈ ^{∘} I,

||(λ + A_{p})^{−1}||_{p→pj} ⩽ c_{p}(λ − ω_{p})^{− \frac{1}{p}} (λ > ω_{p}).

In [6], an inequality similar to (4.2) was proved only for |u|^{\frac{p}{2}} in place of |u|^{\frac{p}{2}} sgn u.

Corollary 4.4. Let the assumptions and notation be as in Theorem 4.2, p ∈ I. 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) and U_{n} → 0 a.e. (n → ∞). Then τ + U_{n} is closable, the analytic semigroup T_{U_{n},2} associated with \overline{τ + U_{n}} extrapolates to a C_{0}-semigroup T_{U_{n},p} on L_{p}, and T_{U_{n},p} = (T_{p})_{U_{n}} → T_{p} as n → ∞.

Proof. Let W := W_{1} + W_{2} + 2V^{−}. Then τ + W is a closed sectorial form, by Proposition 4.1(a). Since τ + U_{n} + W is closed, the form τ + U_{n} is closable by Lemma 3.5. By Lemma 3.13, U_{n} is (τ_{a} + W)-regular and hence (τ + W)-regular. By Proposition 3.11, U_{n} is T_{p}-regular and \overline{τ + U_{n}} ↔ (T_{p})_{U_{n}}, i.e., T_{U_{n},2} and (T_{p})_{U_{n}} are consistent. Now, by [18, Cor. 3.6] we conclude that (T_{p})_{U_{n}} → T_{p} as n → ∞ since U_{0} is T_{p}-regular.

As a direct consequence of Theorem 4.2 we obtain a more explicit version of that theorem.

Corollary 4.5. 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 4.2, 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 3.13, we deduce that τ_{p} ⩾ −(\frac{2}{p}B_{1} + \frac{2}{p′} B_{2} + G) for all p ∈ I_{0}. Thus, Theorem 4.2 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}.

For the remainder of the section, we are concerned with the case b_{2} = 0, V ⩾ 0,

−⟨∇u, b_{1}u⟩ ⩽ (βτ_{a} + V + ω)(u) (0 ⩽ u ∈ D(τ))

for some β < 2, ω ∈ ℝ. Then τ is associated with a consistent family of positive C_{0}-semigroups T_{p} on L_{p}, p ⩾ \frac{2}{2−β}, by Theorem 4.2. The semigroups are L_{∞}- contractive, by Remark 4.3(b).

In Corollary 4.4 we have shown that convergence of potentials implies strong convergence of the corresponding semigroups. Here we discuss approximation of the first order terms. For n ∈ ℕ∪{∞}, let b_{n}: Ω → ℝ^{N} be measurable and define τ_{n} by

τ_{n}(u, v) := τ_{a}(u, v) + ⟨∇u, b_{n}v⟩ + ⟨V u, v⟩

on D(τ_{n}) := D(τ_{a}) ∩ Q(b_{n}^{⊤}a_{s}^{−1}b_{n} + V).

Proposition 4.6. Let (a) hold and assume that b_{n} → b_{∞} a.e., V is τ_{a}-regular, and there exist 0 < β < 2, ω ∈ ℝ, 0 ⩽ U_{0} ∈ L_{1} + L_{∞} such that, for all n ∈ ℕ ∪ {∞}, we have b_{n}^{⊤}a_{s}^{−1}b_{n} ⩽ U_{0} and

−⟨∇u, b_{n}u⟩ ⩽ (βτ_{a} + V + ω)(u) (0 ⩽ u ∈ D(τ_{n})).

Then, for all p ⩾ \frac{2}{2−β}, τ_{n} ↔ T_{p}^{(n)} on L_{p} (n ∈ ℕ ∪ \{∞\}), and T_{p}^{(n)} → T_{p}^{(∞)} as n → ∞.

For the proof of the proposition, we need the following elementary form convergence result which was proved in [16, Thm. A.1] for symmetric forms.

Lemma 4.7. For n ∈ ℕ ∪ {∞}, let τ_{n} be a closed sectorial form in a Hilbert space H, and A_{n} the associated m-sectorial operator. Assume that, for some closed symmetric form 𝔥 ⩾ 1 in H, and some c ⩾ 1, ω ∈ ℝ we have

\frac{1}{c}𝔥 ⩽ Re τ_{n} + ω ⩽ c𝔥 (n ∈ ℕ ∪ \{∞\})

and

sup_{𝔥(v)⩽1} |(τ_{∞} − τ_{n})(u, v)| → 0 as n → ∞ (u ∈ D(𝔥)).

Then A_{n} → A_{∞} in the strong resolvent sense.

Proof. Without restriction assume that ω = 0. For all f, g ∈ H,

⟨A_{n}^{−1}f − A_{∞}^{−1}f, g⟩ = (τ_{∞} − τ_{n})(A_{∞}^{−1}f, (A_{n}^{∗})^{−1}g).

For all g ∈ H, n ∈ ℕ we have 𝔥((A_{n}^{∗})^{−1}g) ⩽ c Re τ_{n}((A_{n}^{∗})^{−1}g) ⩽ c^{2}||g||^{2} since ||(A_{n}^{∗})^{−1}|| ⩽ c. Hence

||A_{n}^{−1}f − A_{∞}^{−1}f|| = sup_{||g||⩽1} |⟨A_{n}^{−1}f − A_{∞}^{−1}f, g⟩| ⩽ sup_{𝔥(v)⩽c^{2}} |(τ_{∞} − τ_{n})(A_{∞}^{−1}f, v)| → 0.

Proof of Proposition 4.6. Let q ∈ (1, \frac{2}{2−β}), U := q′U_{0}. Then τ_{q} + U is nonnegative, by Proposition 4.1(a). Recall from Remark 3.7(a) that U is τ_{a}-regular. Let p ⩾ \frac{2}{2−β}. For n ∈ ℕ ∪ \{∞\}, let T_{p}^{(n)} denote the positive C_{0}-semigroup on L_{p} associated with τ_{n}. Let T_{U,2}^{(n)} denote the C_{0}-semigroup on L_{2} associated with the closed sectorial form τ_{n} + U. Since U is τ_{a}-regular, it is T_{p}^{(n)}-regular and (τ_{n} + U) ↔ (T_{p}^{(n)})_{U}, by Corollary 4.4. Thus, (T_{p}^{(n)})_{U} and T_{U,2}^{(n)} are consistent.

We are going to show that T_{U,2}^{(n)} → T_{U,2}^{(∞)} as n → ∞. This will imply that (T_{p}^{(n)})_{U} → (T_{p}^{(∞)})_{U} for all p ⩾ \frac{2}{2−β} since T_{U,2}^{(n)} is L_{∞}- and L_{q}-contractive. Then the assertion follows from Proposition 2.4.

Without restriction U ⩾ 1. Let 𝔥 := τ_{a} + U + V. It is straightforward that, for all n ∈ ℕ ∪ \{∞\}, we have \frac{1}{2}𝔥 ⩽ τ_{n} + U ⩽ 2𝔥. Moreover, for all u, v ∈ D(𝔥),

|(τ_{∞} − τ_{n})(u, v)|^{2} = |⟨∇u, (b_{n} − b_{∞})v⟩|^{2}
⩽ ⟨U^{−1}(b_{n} − b_{∞})^{⊤}a_{s}^{−1}(b_{n} − b_{∞})(∇u)^{∗}a∇u⟩⟨U|v|^{2}⟩.

Therefore,

sup_{𝔥(v)⩽1} |(τ_{∞} − τ_{n})(u, v)| → 0 (u ∈ D(𝔥))

and hence T_{U,2}^{(n)} → T_{U,2}^{(∞)}, by Lemma 4.7. This completes the proof.

Example 4.8. Here we give several examples of applications of Corollary 4.5 to the case b_{2} = 0, V = 0.

(i) Assume W_{1} ⩽ β^{2} Re τ_{a} + B for some 0 < β < 2, B ⩾ 0, in the sense of quadratic forms on L_{2}. Then, by Euclid’s inequality,

|⟨b_{1}∇u, u⟩| ⩽ \frac{1}{2β} ||W_{1}^{1/2}u||_{2}^{2} + \frac{β}{2} ||a_{s}^{1/2}∇u||_{2}^{2} ⩽ β Re τ_{a}(u) + \frac{B}{2β} ||u||_{2}^{2}.

Hence, by Corollary 4.5, τ is associated with a family of consistent positive quasicontractive C_{0}-semigroups T_{p} on L_{p} with growth bound less or equal \frac{B}{pβ}, for all p ⩾ \frac{2}{2−β}. If β < 1 then τ sectorial and closed. In this case [6, Thm. 1], with use of [4, Thm. VI.2.1], associates τ with a family of consistent analytic quasicontractive C_{0}-semigroups on L_{p}, p ⩾ 2, which coincide with T_{p}.

In [6, Thm. 6], under the additional condition that W_{1} ∈ L_{1} + L_{∞}, τ was associated with a family of consistent C_{0}-semigroups on the same interval of the L_{p}-scale, by approximation of b by bounded vector fields in such a way that the corresponding semigroups converge in L_{p}. Proposition 4.6 shows that the limiting semigroup does not depend on the choice of the approximating sequence. This answers a question posed by V. Liskevich in a remark to [6, Thm. 6]. Moreover, it follows from Proposition 4.6 that the semigroup constructed in [6] coincides with the one constructed in Theorem 4.2.

(ii) Let N ⩾ 2, Ω = ℝ^{N}, a(x) = id, D(τ_{a}) = H^{1}. Let (e_{j})_{j=1}^{N} be the canonical orthonormal basis in ℝ^{N}, (x_{n})_{n=1}^{∞} = ℚ^{N}, (c_{n})_{n=1}^{∞} ⊆ (0, ∞) be such that the potential U(x) = ∑_{n} c_{n}^{2}|x − x_{n}|^{−n} is τ_{a}-regular (see [15] for details of the construction). Let (β_{n})_{n∈ℕ} ⊆ ℝ \ {0} be such that

|β|^{2} := ∑_{n} β_{n}^{2} < ∞

Let b_{1} := ∑_{n=1}^{∞} b_{1n}, where

b_{1n}(x) = c_{n}|x − x_{n}|^{− \frac{n}{2}} β_{n}(\frac{∂|x−x_{n}|}{∂x_{1}}e_{2} − \frac{∂|x−x_{n}|}{∂x_{2}}e_{1}).

We show that ⟨b_{1n}∇u, u⟩ = 0 for all n ∈ ℕ, u ∈ H^{1} ∩ Q(|b_{1n}|^{2}). For u ∈ C_{c}^{1}(ℝ^{N} \ {x_{n}}), the equality follows by integration by parts. For general u ∈ H^{1} ∩ Q(|b_{1n}|^{2}), it then follows from the fact that C_{c}^{1}(ℝ^{N} \ {x_{n}}) is dense in H^{1}∩Q(|b_{1n}|^{2}) and that the form (u, v) ↦ ⟨b_{1n}∇u, v⟩ is bounded on H^{1}∩Q(|b_{1n}|^{2}).

The drift b_{1} is nowhere integrable on ℝ^{N}. However, by the Cauchy-Schwarz inequality,

|b_{1}|(x) ⩽ ∑_{n=1}^{∞} |b_{1n}|(x) ⩽ ∑_{n=1}^{∞} c_{n}|x − x_{n}|^{− \frac{n}{2}} · 2|β_{n}| ⩽ 2|β|U^{\frac{1}{2}} (x).

Hence W_{1} = |b_{1}|^{2} is τ_{a}-regular and ⟨b_{1}∇u, u⟩ = 0 for all u ∈ H^{1} ∩ Q(U).

Thus, τ is associated with a consistent family of positive contractive C_{0}- semigroups T_{p} on L_{p}, p ⩾ 1.

(iii) Let N ⩾ 2, Ω = ℝ^{N}, a(x) = id, D(τ_{a}) = H^{1}. Let b_{1}(x) = cx|x|^{α} for some c, α ∈ ℝ. Then |b_{1}|^{2} is τ_{a}-regular. Moreover,

−⟨b_{1}∇u, u⟩ = \frac{c}{2}(N + α)⟨r^{α}u^{2}⟩ (0 ⩽ u ∈ C_{c}^{1}(ℝ^{N} \ 0)).

Hence, if c(N + α) ⩽ 0 then τ is associated with a consistent family of positive contractive C_{0}-semigroups on L_{p}, p ⩾ 1. If N ⩾ 3 we can use the Hardy inequality ||\frac{u}{r}||_{2}^{2} ⩽ \frac{4}{(N−2)^{2}} ||∇u||_{2}^{2} to treat the case c(N +α) > 0 with −2 ⩽ α ⩽ 0. For α = −2, τ is associated with a quasi-contractive C_{0}-semigroup on some L_{p} if (and only if, see Remark 6.5 below) c < N − 2, and then τ is associated with a consistent family of positive contractive C_{0}-semigroups on L_{p}, p ⩾ \frac{N−2}{N−2−c}. For α ∈ (−2, 0], we use the fact that r^{α} ⩽ εr^{−2} + C_{α,ε} for all r, ε > 0 with some constant C_{α,ε} to conclude that in this case τ is associated with a consistent family of positive quasi-contractive C_{0}-semigroups on L_{p}, p > 1. If α = 0 then the semigroup extrapolates also to L_{1}.

5 Proof of the main theorem

We separate the core of the proof of Theorem 4.2 into a lemma. Let 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}, v_{p}(u) := u|u|^{\frac{p}{2} −1} and w_{p}(u) := u|u|^{p−2}.

Lemma 5.1. 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 (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}⟩  (5.1)

for all u ∈ D(τ), n ∈ ℕ.

(a) Then τ is associated with a positive C_{0}-semigroup T_{p}(t) = e^{−A_{p}t} on L_{p} with ||T_{p}(t)|| ⩽ e^{ωt} (t ⩾ 0), and for all u ∈ D(A_{p}) we have v_{p}(u) ∈ D(𝔥) and

Re⟨A_{p}u, w_{p}(u)⟩ ⩾ 𝔥(v_{p}(u)).  (5.2)

(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)  (5.3)

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. The proof is divided into three steps. In step (i) we consider the m-sectorial operator A_{0} in L_{2}, associated with τ + U_{0}, and show that e^{−A_{0}t} extrapolates to a contractive C_{0}-semigroup T_{0,p}(t) = e^{−A_{0,p}t} on L_{p}. In step (ii) we show that −U_{0} is T_{0,p}-admissible and (T_{0,p})_{−U_{0}} is a contractive C_{0}-semigroup. This proves the first assertion of (a). The second assertion is proved in step (iii).

(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 (5.1) 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} (5.4)

Observe that |w_{n,p}|^{p′} = |u|^{p′}u_{n,p}^{2p′} ⩽ |v_{n,p}|^{2}. Hence ||w_{n,p}||_{p′} ⩽ ||v_{n,p}||2^{\frac{2}{p′}}, and from

estimate (5.4) we obtain that

||v_{n,p}||2^{\frac{2}{p}} ⩽ \frac{1}{λ}||f||_{p} (n ∈ ℕ).

Since |v_{n,p}| ↑ |v_{p}(u)| we conclude by the Beppo Levi theorem that v_{p}(u) ∈ L_{2}, and

||(λ + A_{0})^{−1}f||_{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 Lebesgue’s dominated convergence theorem, v_{n,p} → v_{p}(u) in L_{2} and w_{n,p} → w_{p}(u) in L_{p′}. Further, A_{0}u = f − λu ∈ L_{p}. From estimate (5.4) 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 the Beppo Levi theorem, (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)⟩.  (5.5)

So far we have proved inequality (5.5) 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′}. From (5.5) we conclude that

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)⟩.

The lower semicontinuity of 𝔥 + U_{0} implies that (5.5) holds for all u ∈ D(A_{0,p}).

For m ∈ ℕ, let A_{m} := A_{0,p} − U_{0} ∧ m. Then A_{m} is a closed operator and, by (5.5), Re⟨A_{m}u, w_{p}(u)⟩ ⩾ 0 for all u ∈ D(A_{m}) = D(A_{0,p}). By the Lumer-Phillips theorem, e^{−A_{m}t} = (T_{0,p})_{−U_{0}∧m}(t) is a contractive C_{0}-semigroup on L_{p} and, by [18, Prop. 2.2] (see Definition 2.1(b)), we conclude that −U_{0} is T_{0,p}- admissible and that T_{p} := (T_{0,p})_{−U_{0}} is a contractive C_{0}-semigroup on L_{p}.

(iii) Let −A_{p} be the generator of T_{p}. By (ii), A_{m} → A_{p} in the strong resolvent sense. Let u ∈ D(A_{p}). Then u^{(m)} := (1 + A_{m})^{−1}(1 + A_{p})u → 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}. Furthermore, v_{p}(u^{(m)}) → v_{p}(u) in L_{2} and w_{p}(u^{(m)}) → w_{p}(u) in L_{p′} as m → ∞. Hence, estimate (5.5) yields

lim inf_{m} 𝔥(v_{p}(u^{(m)})) ⩽ lim⟨A_{m}u^{(m)}, w_{p}(u^{(m)})⟩ = ⟨A_{p}u, w_{p}(u)⟩.

The lower semicontinuity of 𝔥 implies (5.2).

(b) Let u ∈ D(A_{0}) ∩ D(A_{0,p}). Then, since u\overline{w}_{n,p} is real,

Im⟨A_{m}u, w_{n,p}⟩ = Im⟨(A_{0} − U_{0} ∧ m)u, w_{n,p}⟩ = Im τ(u, w_{n,p}).

By (5.5) 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 Lebesgue’s dominated convergence theorem. By (5.3) 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}) and u^{(m)} be as in the beginning of step (iii). Then

|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.

For the application of Lemma 5.1 in the proof of Theorem 4.2 we need to compute the gradient of v_{n,p} and w_{n,p}.

Lemma 5.2. For α ∈ ℝ, r > 0, z ∈ ℂ, denote z_{α,r} := |z|^{α} ∧ r if α ≠ 0 and z_{0,r} := 1 ∧ r. Let φ: ℂ → ℂ, φ(z) = zz_{α,r}. Then, for all (complex valued) u ∈ W_{1,loc}^{1}, v = φ ∘ u ∈ W_{1,loc}^{1} and

∇v = u_{α,r}(∇u + α11_{{|u|^{α}<r}} sgn u∇|u|).

Proof. It is easy to see that φ is a Lipschitz continuous function. So v = φ∘u is in W_{1,loc}^{1}. If α ∉ (0, 1) then the function [0, ∞) ∋ t ↦ t^{α} ∧ r is Lipschitz continuous too, hence u_{α,r} ∈ W_{1,loc}^{1}, ∇u_{α,r} = α11_{{|u|^{α}<r}}|u|^{α−1}∇|u| and the second statement of the lemma follows from the general product rule.

Let now 0 < α < 1. We denote z_{δ,α,r} = (|z| + δ)^{α} ∧ r and approximate φ with the functions φ_{δ}, φ_{δ}(z) := zz_{δ,α,r}. The function [0, ∞) ∋ t ↦ (t + δ)^{α} ∧ r is Lipschitz continuous and

∇u_{δ,α,r} = α11_{{(|u|+δ)^{α}<r}}(|u| + δ)^{α−1}∇|u|.

So, by the general product rule,

∇(φ_{δ} ∘ u) = u_{δ,α,r}(∇u + α\frac{u}{|u|+δ}11_{\{(|u|+δ)^{α}<r\}}∇|u|).

Finally, φ_{δ} ∘ u → φ ∘ u and ∇(φ_{δ} ∘ u) → u_{α,r}(∇u + α11_{{|u|^{α}<r}} sgn u∇|u|) in L_{1,loc} by Lebesgue’s dominated convergence theorem, which implies the assertion.

Proof of Theorem 4.2. Let p ∈ I, i.e., τ_{p} ⩾ −ω_{p}. Let U_{0} := W_{1} + W_{2} + 2V^{−}. By Proposition 4.1(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}⟨11_{n}(W_{1} + W_{2})|v_{n,p}|^{2}⟩,  (5.6)

where 11_{n} is the indicator of the set \{x; |u|^{\frac{p−2}{2}} ⩾ n\}. Applying Lemma 5.1(a) with 𝔥 = \overline{τ_{p}} and U_{n} = \frac{1}{2}11_{n}(W_{1} + W_{2}) (n ∈ ℕ), we obtain all the assertions of Theorem 4.2 except for the analyticity of T_{p}.

Let now assumption (4.3) hold for some p ∈ ^{∘} I. Then it holds for all p ∈ ^{∘} I, by Proposition 4.1(c). 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}⟩| (5.7)

which is also shown at the end of the proof. The first term in the right hand side of (5.7) can be estimated by α Re τ_{a}(v_{n,p}), due to assumption (a). Thus, by (4.3) 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 4.1(b) we have Re τ_{a}(v_{n,p}) ⩽ C(τ_{p} + ω̃_{1})(v_{n,p}) for some ω̃_{1} ∈ ℝ, C > 0 depending on p. Moreover, τ_{p}(v_{n,p}) ⩽ (Re τ + U_{n})(u, w_{n,p}) by (5.6). 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 5.1(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. Let U_{0} be as above. Then τ̃ := τ + U_{0} 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′} ⟨u, b_{2}∇u⟩ + \frac{1}{p}(2⟨b_{1}∇u, 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⟨b_{1}∇u, u⟩ + ⟨V u^{2}⟩ ⩾ −ω_{1}||u||_{2}^{2}

to the third term in 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 4.2 applied to τ̃ implies: 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})). (5.8)

In particular, ||T_{0,p}(t)||_{p→p} ⩽ e^{\frac{ω_{1}}{p} t} for all t ⩾ 0, 1 < p ⩽ 4. Since T_{0} is a positive C_{0}-semigroup, [19] 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 (5.8) 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 2.3(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 [17, Prop. A.2]. Hence

sup_{n∈ℕ}||(T_{0,1})_{−U_{0}∧n}(t)||_{1→1} ⩽ e^{ω_{1}t} (t ⩾ 0).

Finally, [18, 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 (5.6) and (5.7). Let 11_{n}^{c} := 1 − 11_{n}, i.e., the indicator of the set \{x; |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 5.2 implies that

∇v_{n} = u_{n}(∇u + \frac{p−2}{2}11_{n}^{c} sgn u∇|u|) = sgn u(u_{n} sgn \overline{u}∇u + \frac{p−2}{2}11_{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}11_{n}^{c}φ_{n} = (\frac{p}{2}11_{n}^{c} + 11_{n})φ_{n} + iψ_{n}.

In the same way, with ρ_{n} = (p − 1)11_{n}^{c} + 11_{n}, we have

∇\overline{w}_{n} = u_{n}^{2}(∇\overline{u} + (p − 2)11_{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}),
\begin{matrix}a∇v_{n} · ∇\overline{v}_{n} & = a(sgn \overline{u}∇v_{n}) · (sgn u∇\overline{v}_{n}) \\ = (\frac{p^{2}}{4} 11_{n}^{c} + 11_{n})a_{s}φ_{n} · φ_{n} + a_{s}ψ_{n} · ψ_{n} + i(a − a_{s})ψ_{n} · (p11_{n}^{c} + 211_{n})φ_{n}. \end{matrix} (5.9)

Therefore Re a∇u · ∇\overline{w}_{n} = ((p − 1)11_{n}^{c} + 11_{n})a_{s}φ_{n} · φ_{n} + a_{s}ψ_{n} · ψ_{n}. Note that \frac{4}{pp′} \frac{p^{2}}{4} = p − 1. Hence, we obtain

Re τ_{a}(u, w_{n}) = \frac{4}{pp′} Re τ_{a}(v_{n}) + (1 − \frac{4}{pp′})⟨11_{n}a_{s}φ_{n} · φ_{n} + a_{s}ψ_{n} · ψ_{n}⟩.

For the first order terms we compute

\overline{v}_{n}∇v_{n} = |v_{n}|((\frac{p}{2}11_{n}^{c} + 11_{n})φ_{n} + iψ_{n}),
\overline{w}_{n}∇u = |v_{n}|u_{n} sgn \overline{u}∇u = |v_{n}|(φ_{n} + iψ_{n}),  (5.10)
u∇\overline{w}_{n} = |v_{n}|(ρ_{n}φ_{n} − iψ_{n}).

Thus, Re \overline{v}_{n}∇v_{n} = |v_{n}|(\frac{p}{2}11_{n}^{c} + 11_{n})φ_{n}. We obtain that

Re \overline{w}_{n}∇u = |v_{n}|φ_{n} = \frac{2}{p} Re(\overline{v}_{n}∇v_{n}) + (1 − \frac{2}{p})11_{n}|v_{n}|φ_{n}

and, since \frac{2}{p′} \frac{p}{2} = p − 1,

Re u∇\overline{w}_{n} = ((p − 1)11_{n}^{c} + 11_{n})|v_{n}|φ_{n} = \frac{2}{p′} Re(\overline{v}_{n}∇v_{n}) + (1 − \frac{2}{p′})11_{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}⟨11_{n}a_{s}φ_{n} · φ_{n} + a_{s}ψ_{n} · ψ_{n}⟩ + ε_{p}⟨11_{n}(b_{1} + b_{2})|v_{n}| · φ_{n}⟩.

This implies (5.6) since ε_{p}11_{n}|(b_{1} +b_{2})v_{n} ·φ_{n}| ⩽ ε_{p}^{2}11_{n}a_{s}φ_{n} ·φ_{n} + \frac{1}{2}11_{n}(W_{1} +W_{2})|v_{n}|^{2}, by Euclid’s inequality.

To prove (5.7), we first compute Im τ_{a}(u, w_{n}).

Im(a∇u · ∇\overline{w}_{n}) = ((p − 1)11_{n}^{c} + 11_{n})aψ_{n} · φ_{n} − aφ_{n} · ψ_{n}
= (p − 2)11_{n}^{c}a_{s}ψ_{n} · φ_{n} + (p11_{n}^{c} + 211_{n})(a − a_{s})ψ_{n} · φ_{n}.

The second term in the right hand side equals Im(a∇v_{n} · ∇\overline{v}_{n}), by (5.9). The first term we estimate, using Euclid’s inequality and (5.9), as follows:

|(p − 2)11_{n}^{c}a_{s}ψ_{n} · φ_{n}| ⩽ |p − 2|11_{n}^{c}(\frac{p}{4}a_{s}φ_{n} · φ_{n} + \frac{1}{p}a_{s}ψ_{n} · ψ_{n})

= |1 − \frac{2}{p}|11_{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 (5.10),

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 (5.7) follows.

6 Sharpness of the result

In this section we show that, under some conditions additional to (a) and (bV), if τ ↔ T_{p} on L_{p} for some p ∈ (1, ∞), with ||T_{p}(t)|| ⩽ e^{ω_{p}t} for some ω_{p} ∈ ℝ, then estimate (1.3) holds.

Lemma 6.1. Let 1 < p < ∞, T_{p} a positive contractive C_{0}-semigroup on L_{p}. Let U ⩾ 0 be a T_{p}-admissible potential, −A_{U} the generator of (T_{p})_{U}. Then

Re⟨A_{U}u, u|u|^{p−2}⟩ ⩾ ⟨U|u|^{p}⟩ (u ∈ D(A_{U})).

Proof. Let −A be the generator of T_{p}. For m ∈ ℕ let U_{m} = U ∧m. Let u ∈ D(A_{U}) and u_{m} := (1 + A + U_{m})^{−1}(1 + A_{U})u. Since A is accretive, we have

Re⟨(1 + A_{U})u, u_{m}|u_{m}|^{p−2}⟩ = Re⟨(1 + A + U_{m})u_{m}, u_{m}|u_{m}|^{p−2}⟩ ⩾ ⟨(1 + U_{m})|u_{m}|^{p}⟩.

Since u_{m} → u in L_{p} and U_{m} ↑ U, we complete the proof by an application of Fatou’s lemma.

The following theorem is the main part of our sharpness result.

Theorem 6.2. Let (a), (bV) hold and assume that τ ↔ T_{p} on L_{p} for some p ⩾ 2, with ||T_{p}(t)|| ⩽ e^{ω_{p}t} (t ⩾ 0) for some ω_{p} ∈ ℝ. If there exists a τ_{a}-regular potential U ⩾ 0 such that ||(T_{p})_{U}(t)||_{∞→∞} ⩽ C for all t ⩾ 0 then estimate (1.3) holds.

If ⟨∇u, b_{2}u⟩ ⩽ ω||u||_{2}^{2} (u ∈ D(τ)) and U ⩾ V^{−} + ω, then ||(T_{p})_{U}(t)||_{∞→∞} ⩽ 1 for all t ⩾ 0, by Remark 4.3(b) and Proposition 3.11.

The proof of Theorem 6.2 is based on the following lemma.

Lemma 6.3. Let (M, μ) be a measure space, 𝔥 a Dirichlet form in L_{2}(μ) and r ⩾ 1.

(a) Then D_{1} := {0 ⩽ u ∈ D(𝔥) ∩ L_{∞}(μ); u^{1/r} ∈ D(𝔥)} is dense in D(𝔥)_{+}, the set of positive elements of 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^{−At}||_{∞→∞} ⩽ 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 the functions φ_{n} are Lipschitz continuous with constant r, φ_{n}^{1/r} are Lipschitz continuous and that φ_{n}(s) → s (s ⩾ 0) as n → ∞. For u ∈ D(𝔥)_{+} we conclude that φ_{n}(u) ∈ D_{1}, and from [1, 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}. Let u ∈ D_{1} and v := u^{1/r}. Then v ∈ D(𝔥) ∩ L_{∞}(μ). By [9, 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 assumption on A implies that v_{λ} ∈ D(A) ∩ L_{∞} and ||v_{λ}||_{∞} ⩽ 2C||v||_{∞} for large λ. Moreover, we have Av_{λ} = λ(v − v_{λ}) ∈ L_{∞}. Therefore, v_{λ}^{r} ∈ D_{2} and, by [1, Théorème 10], v_{λ}^{r} → v^{r} = u in D(𝔥) as λ → ∞.

Proof of Theorem 6.2. Without restriction assume U ⩾ U_{0} := W_{1} +W_{2} +2|V | (see Lemma 3.13). 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 3.13, it therefore suffices to show τ_{p}(u) ⩾ −ω_{p}||u||_{2}^{2} for all u ∈ D(τ_{a} + U). Since τ_{p} fulfills the first Beurling-Deny criterion we can restrict ourselves to u ⩾ 0.

By Proposition 4.1(a), τ + U is a closed sectorial form in L_{2}. Let A_{U} be the m-sectorial operator in L_{2} associated with τ + U. Then the assumptions of Lemma 6.3(b) are fulfilled with 𝔥 = τ_{a} + U, 𝔥_{1} = τ + U, A = A_{U} since e^{−A_{U} t} and (T_{p})_{U} are consistent by Corollary 4.4. Below we show that

τ_{p}(u^{\frac{p}{2}} ) ⩾ −ω_{p}||u^{\frac{p}{2}} ||_{2}^{2}  (6.1)

for all 0 ⩽ u ∈ D(A_{U}) ∩ L_{∞} with A_{U}u ∈ L_{∞}. Then, an application of Lemma 6.3(b) shows that τ_{p}(u) ⩾ −ω_{p}||u||_{2}^{2} for all 0 ⩽ u ∈ D(τ_{a} + U), and the proof is complete.

So, let 0 ⩽ u ∈ D(A_{U}) ∩ L_{∞} with 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 3) and thus, by the definition of A_{U}, (τ_{p} + U)(u^{\frac{p}{2}} ) = ⟨A_{U}u, u^{p−1}⟩.

Since e^{−A_{U} t} and e^{−A_{p,U} t} := (T_{p})_{U} are consistent and u, A_{U}u ∈ L_{2} ∩ L_{∞} ⊆ L_{p}, we obtain u ∈ D(A_{p,U}) and A_{p,U}u = A_{U}u. By Lemma 6.1 we infer that

(τ_{p} + U)(u^{\frac{p}{2}} ) = ⟨A_{p,U}u, u^{p−1}⟩ ⩾ ⟨(U − ω_{p})u^{p}⟩,

i.e., (6.1) holds.

By Proposition 3.12 we easily obtain the following corollary.

Corollary 6.4. Let (a), (bV) hold and assume that, for some τ_{a}-regular potential U ⩾ 0, the form τ + U is sectorial and closable and the associated semigroup T_{U} satisfies ||T_{U}(t)||_{1→1} ⩽ C, ||T_{U}(t)||_{∞→∞} ⩽ C (t ⩾ 0). If τ ↔ T_{p} on L_{p} for some p ∈ (1, ∞), with ||T_{p}(t)|| ⩽ e^{ω_{p}t} (t ⩾ 0) for some ω_{p} ∈ ℝ, then estimate (1.3) holds.

Remark 6.5. The previous result 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.

Acknowledgment

Parts of this paper were written while the second author was visiting the School of Mathematics, University of Bristol. It is a pleasure to thank the department for the hospitality. The authors would also like to thank Vitali Liskevich and Jürgen Voigt for enlightening discussions. Support of EPSRC through Grant GR/L76891 and of the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

References

[1] A. Ancona, Continuité des contractions dans les espaces de Dirichlet., in: Sem. Th. Potentiel. Paris no.2. Lecture Notes in Math. 563, 1–26, Springer Verlag Berlin, 1976.

[2] A. Beliy and Yu. Semenov, On the L^{p}-theory of the Schrödinger semigroups, II, Siberian Math. J. 31(4) (1990), 16–26.

[3] E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1989.

[4] T. Kato, Perturbation Theory of Linear Operators, 2nd edition, Springer-Verlag, Berlin-Heidelberg-New York, 1980.

[5] V. Kovalenko and Yu. Semenov, C_{0}-semigroups in L^{p}(ℝ^{d}) and Ĉ(ℝ^{d}) spaces generated by the differential expression Δ + b · ∇, Theory Probab. Appl. 35 (1990), 443–453.

[6] V. Liskevich, On C_{0}-semigroups generated by elliptic second order differential expressions on L^{p}-spaces, Differential Integral Equations 9 (1996), no.4, 811–826.

[7] V. Liskevich and A. Manavi, Dominated semigroups with singular complex potentials, J. Funct. Anal. 151 (1997), 281–305.

[8] V. Liskevich, Z. Sobol and H. Vogt, On L_{p}-theory of C_{0}-semigroups associated with second order elliptic operators. II, preprint.

[9] Z. Ma and M. Röckner, Introduction to the theory of (non-symmetric) Dirichlet forms, Springer-Verlag, Berlin-Heidelberg, 1992.

[10] A. Manavi, Zur Störung von dominierten C_{0}-Halbgruppen auf Banachfunktionenräumen mit ordnungsstetiger Norm und sektoriellen Formen mit singulären komplexen Potentialen, doctoral dissertation, 2001.

[11] E. M. Ouhabaz, L^{∞}-contractivity of semigroups generated by sectorial forms, J. London Math. Soc. (2) 46 (1992), 529–542.

[12] M. Perelmuter and Yu. Semenov, On decoupling of finite singularities in the scattering theory for the Schrödinger operators with a magnetic field, J. Math. Phys. 22(3) (1981), 521–533.

[13] M. Röckner and N. Wielens, Dirichlet forms - closability and change of speed measure, in Infinite-dimensional analysis and stochastic processes (Bielefeld, 1983), Res. Notes in Math. 124, Pitman, Boston, Mass.-London, 1985, 119–144.

[14] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447–526.

[15] P. Stollmann and J. Voigt, A regular potential which is nowere in L_{1}, Letters in Math. Phys. 9 (1985), 227–230.

[16] P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures Potential Anal. 5 (1996), 109–138.

[17] J. Voigt, Absorption semigroups, their generators and Schrödinger semigroups, J. Funct. Anal. 67 (1986), 167–205.

[18] J. Voigt, Absorption semigroups, J. Operator Theory 20 (1988), 117–131.

[19] J. Voigt, One-parameter semigroups acting simultaneously on different L^{p} spaces, Bull. Soc. Roy. Sci. Liège 61 (1992), 465–470.