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

Vitali Liskevich

School of Mathematics University of Bristol Bristol BS8 1TW, UK

Zeev Sobol

School of Mathematics University of Bristol Bristol BS8 1TW, UK

Hendrik Vogt

Institut für Analysis Fachrichtung Mathematik Technische Universität Dresden 01062 Dresden, Germany

Abstract

We study positive C_{0}-semigroups on L_{p} associated with second order uniformly elliptic divergence type operators with singular lower order terms, subject to a wide class of boundary conditions. We obtain an interval (p_{min}, p_{max}) in the L_{p}-scale where these semigroups can be defined, including the case 2 ∉ (p_{min}, p_{max}). We present an example showing that the result is optimal. We also show that the semigroups are analytic with angles of analyticity and spectra of the generators independent of p, for the whole range of p where the semigroups are defined.

1 Introduction and main results

In this paper we continue to study the L_{p}-theory of second order elliptic differential operators on an open set Ω ⊆ ℝ^{N}, N ⩾ 3, 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. In [24] a quasi-contractive C_{0}-semigroup on L_{p} := L_{p}(Ω) is constructed, whose generator is associated with L. In this paper we study the case of uniformly elliptic operators and show that, under some additional restrictions, the range of L_{p}-spaces in which one can associate a C_{0}-semigroup with L, can be extended beyond the interval of quasi-contractivity. We also prove that the consistent semigroups associated with L on L_{p} are analytic with angles of analyticity and spectra of the generators independent of p.

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) responding 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.)

Our main interest lies in the case when the semigroup associated with L can be defined on L_{p} for p from a proper subinterval of [1, ∞). This case of the L_{p}-theory of second order elliptic operators has been extensively studied [2, 5, 12, 15, 18, 19, 20, 21]. However, most of the results are related to sectorial forms (especially to symmetric forms bounded below) and quasi-contractive semigroups. In [24] a general method of constructing positive C_{0}-semigroups on L_{p} corresponding to sesquilinear (not necessarily sectorial) forms in L_{2} has been developed, and a precise condition for quasi-contractivity has been established.

It was first observed in [11] that the Schrödinger semigroup with L_{N/2,weak}- potential can be defined on L_{p} for certain p outside of the interval of quasicontractivity. In [21] this result was extended to uniformly elliptic second order divergence type operators in ℝ^{N} perturbed by a form bounded potential. Here we study a general second order differential expression L for a wide class of boundary conditions.

E.-M. Ouhabaz [17] was the first to establish analyticity of angle \frac{π}{2} in L_{p}(ℝ^{N}), 1 ⩽ p < ∞, for symmetric semigroups satisfying Gaussian upper bounds. E. B. Davies [5] extended this result to a more general setting of metric spaces with polynomial volume growth. In [18] analyticity of angle \frac{π}{2} was first shown for symmetric semigroups that are defined only for p from an interval in [1, ∞), under the assumption of certain weighted estimates. In the present paper we prove analogous results for general uniformly elliptic second order operators. The result on p-independence of the spectrum we present here, which is an application of a criterion from [16], generalizes respective results from [8, 19, 20, 21].

The main tool of the present paper is a technique of weighted estimates analogous to that used in [19, 5, 18]. For further development of this technique with applications to L_{p}-theory we refer the reader to [14, 26].

We recall from [24] the following qualitative assumptions on the form τ.

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

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

for some α ⩾ 0, i.e., a is uniformly sectorial (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}. 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., Q(W_{j})∩ D(τ_{a}) and Q(|V |) ∩ D(τ_{a}) are dense in D(τ_{a}).

(Q(V) denotes the form domain of the multiplication operator V in L_{2}.)

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

As shown in [24], D(τ) is dense in D(τ_{a}), and the form τ + U_{0} − U_{0} ∧ m is sectorial and closed for all U_{0} ⩾ W_{1} + W_{2} + 2V^{−} and m ∈ ℕ.

In order to formulate the main result from [24] we need to introduce the following quadratic forms:

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

on D(τ_{p}) := D(τ) (1 ⩽ p < ∞).

The construction of the quasi-contractive C_{0}-semigroup on L_{p}, corresponding to the form τ, is given in the following theorem which is the main result in [24] (see [24, Thm. 1.1 and Cor. 4.4]).

Theorem 1.1. Let assumptions (a) and (bV) be fulfilled. Let U_{0} ⩾ W_{1} + W_{2} + 2V^{−} be such that Q(U_{0}) ∩ D(τ_{a}) is dense in D(τ_{a}), and T_{0} = T_{0,2} the C_{0}-semi-group associated with the form τ + U_{0} on L_{2}. Let I be the set of all p ∈ [1, ∞) such that τ_{p} ⩾ −ω_{p} for some ω_{p} ∈ ℝ.

(i) Then I is an interval in [1, ∞), and T_{0} extrapolates to a positive C_{0}-semi-group 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 positive 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, and for 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).  (1.2)

(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))  (1.3)

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

As shown in [24], the semigroup T_{p} does not depend on the choice of U_{0}. We say that the semigroup T_{p} is associated with the form τ.

In the rest of the paper we assume that a ∈ L_{∞}. Moreover, we make the following assumption:

(BC) For all φ ∈ W_{∞}^{1}, if u ∈ D(τ_{a}) then φu ∈ D(τ_{a}).

The above assumption is a restriction on the type of boundary conditions. It holds in the case of Neumann boundary conditions, i.e. τ_{a} = τ_{N}, and one can easily see that it is also satisfied if D(τ_{a}) is an ideal of D(τ_{N}) (u ∈ D(τ_{a}), v ∈ D(τ_{N}) and |v| ⩽ |u| imply that v ∈ D(τ_{a})). In particular, it is satisfied in case of Dirichlet boundary conditions. However, (BC) does not hold for periodic type boundary conditions.

Now we are ready to formulate the main result of this paper.

Theorem 1.2. Let (a), (bV) and (BC) hold, and let the interior ^{∘} I =: (p_{−}, p_{+}) of the interval I defined in Theorem 1.1 be non-empty. Assume that

(i) the matrix a is uniformly elliptic, i.e., there exists σ ⩾ 1 such that

σ^{−1} id ⩽ a_{s} ⩽ σ id;

(ii) for some p ∈ ^{∘} I, (1.3) holds and, for some C ⩾ 0,

|⟨(b_{1} + b_{2})|u|^{2}⟩| ⩽ C\sqrt{(τ_{p} + C)(u)}||u||_{2} (u ∈ D(τ));  (1.4)

(iii) D(τ_{a}) ⊆ L_{\frac{2N}{N−2}} .

For q ∈ I, let T_{q} be the semigroup constructed in Theorem 1.1. Let p_{max} := \frac{N}{N−2}p_{+}, p_{min} := (\frac{N}{N−2}p′_{−})′.

Then T_{q}(t)↾_{L_{∞,c}} extends to an analytic C_{0}-semigroup on L_{p} for all p ∈ (p_{min}, p_{max}). The sector of analyticity and the spectrum of the generators are p-independent. For p_{min} < p < q < p_{max}, there exist constants c_{1}, c_{2} > 0 such that

||T_{p}(t)||_{p→q} ⩽ c_{1}t^{− \frac{N}{2} (\frac{1}{p} − \frac{1}{q})}e^{c_{2}t}.  (1.5)

In case 1 ∈ I the assertions hold for all p ∈ [1, p_{max}).

Remarks. 1. By [24, Prop. 4.1(b)], condition (1.4) holds in particular if, for some C ⩾ 0,

|⟨(b_{1} + b_{2})|u|^{2}⟩| ⩽ C||u||_{H^{1}}||u||_{2} (u ∈ H^{1}).

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}⟩ .

2. Assumption (iii) of the theorem is in fact the Sobolev imbedding theorem which holds, for example, for Dirichlet boundary conditions or if the domain Ω satisfies the cone property or the extension property [1].

3. In Section 4 we present an example of a semigroup that cannot be extended to a wider interval in the L_{p}-scale than that obtained in Theorem 1.2. In this sense the result of Theorem 1.2 is sharp. For b_{1} = b_{2} = 0 the interval (p_{min}, p_{max}) was computed in [21].

As a direct consequence of Theorem 1.2 we obtain a variant of that theorem in which the interval (p_{min}, p_{max}) is more explicit.

Corollary 1.3. Let assumptions (a), (bV) and (BC) be fulfilled. Let V_{+}, V_{−} ⩾ 0 be τ_{a}-regular with V_{+} − V_{−} = V, and τ_{+} := Re τ_{a} + V_{+}. Assume that the matrix a is uniformly elliptic, D(τ_{a}) ⊆ L_{\frac{2N}{N−2}} , and

(−1)^{j}⟨b_{j}u, ∇u⟩ ⩽ β_{j}τ_{+}(u) + B_{j}||u||_{2}^{2}, ⟨V_{−}u^{2}⟩ ⩽ γτ_{+}(u) + G||u||_{2}^{2},
⟨|b_{1} + b_{2}|^{2}u^{2}⟩ ⩽ K(τ_{+}(u) + ||u||_{2}^{2})  (1.6)

(0 ⩽ u ∈ D(τ)∩Q(V_{+}), j = 1, 2) for some constants β_{1}, β_{2}, γ ⩾ 0, B_{1}, B_{2}, G, K ∈ ℝ. Let I be the interval defined in Theorem 1.1.

Suppose that (p_{−}, p_{+}) := \{p ∈ [1, ∞); \frac{4}{pp′} − \frac{2}{p}β_{1} − \frac{2}{p′} β_{2} − γ > 0\} ≠ ∅. Then (p_{−}, p_{+}) ⊆ I, and all the assertions of Theorem 1.2 hold with p_{max} := \frac{N}{N−2}p_{+}, p_{min} = (\frac{N}{N−2}p′_{−})′.

Proof. The inclusion holds by [24, Cor. 4.5]. Condition (1.6) implies that assumption (ii) of Theorem 1.2 is fulfilled. Then, by Theorem 1.2, the assertion follows.

The rest of the paper is organized as follows. In Section 2 we present an abstract result on weighted estimates which is a main tool in the proof the main theorem which is given in Section 3. Sharpness of the main result is shown in Section 4. In Section 5 we discuss L_{p}-theory for non-divergence type elliptic operators.

2 Technique of weighted estimates

In this section we are going to show the following theorem which contains an abstract statement needed for the proof of our main result and is useful in some other applications.

Theorem 2.1. Let 1 ⩽ p ⩽ r_{0} ⩽ q ⩽ ∞, T an analytic semigroup of angle θ ∈ (0, \frac{π}{2}] on L_{r_{0}} satisfying

||e^{ξx}T (t)e^{−ξx}||_{p→q} ⩽ Mt^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q})}e^{μ|ξ|^{m}t+ωt} (t > 0, ξ ∈ ℝ^{N})  (2.1)

for some M, μ > 0, m > 1 and ω ∈ ℝ. Then T extrapolates to an analytic semigroup of angle θ on L_{r} for all r ∈ [p, q] \ {∞}, and the spectrum of the generators −A_{r} is independent of r.

This theorem is a generalization of Theorem 2.3 in [9]. There the case p = 1, q = ∞ is treated by showing estimates on the integral kernels of powers of the resolvents (λ + A)^{−1} for λ from some sector. In this case one can use Davies’ trick to show that estimate (2.1) is equivalent to a Gaussian estimate of order m of the integral kernel of the semigroup (cf. [6]).

The main tools needed in the proof of the theorem are Stein interpolation and the following lemma on weighted estimates which is a refinement of Proposition 3.2 from [19].

Lemma 2.2. Let 1 ⩽ p ⩽ q ⩽ ∞, γ > 0. Let B: L_{∞,c} → L_{1,loc} be a linear operator satisfying

||e^{ξx}Be^{−ξx}||_{p→q} ⩽ 1 for all ξ ∈ ℝ^{N} with |ξ| = γ.

Then ||B||_{r→r} ⩽ c_{N}γ^{−N( \frac{1}{p} − \frac{1}{q})} for all r ∈ [p, q], where the constant c_{N} depends only on the dimension N.

Proof. For γ = 1 the lemma is proved in [19], with c_{N} = e^{\sqrt{N}}||(e^{−|k|})_{k}||_{1}. (In fact, there the estimate ||e^{ξx}Be^{−ξx}||_{p→q} ⩽ 1 is assumed for all |ξ| ⩽ 1, but only |ξ| = 1 is used in the proof.) Using a rescaling argument, we now deduce the assertion for general γ.

Define the operator D_{γ} by D_{γ}f(x) := f(γx) for all f: Ω → ℂ and all x ∈ Ω. Then ||D_{γ}f||_{r} = γ^{− \frac{N}{r}} ||f||_{r} for all r ∈ [1, ∞], f ∈ L_{r}. Moreover, D_{γ} ∘e^{ξx} = e^{γξx}∘D_{γ} for all ξ ∈ ℝ^{N}. From the assumption we thus obtain, with B̃ := D_{γ}^{−1}BD_{γ},

||e^{ξx}B̃e^{−ξx}||_{p→q} = ||D_{γ}^{−1}e^{γξx}Be^{−γξx}D_{γ}||_{p→q} ⩽ γ^{−N( \frac{1}{p} − \frac{1}{q})} for all |ξ| = 1.

An application of the lemma in the known case γ = 1 completes the proof.

It should be pointed out that Lemma 2.2 is of particular interest for large γ. Similar results have first been used in [6] and [18], the difference being that there a weighted norm estimate for all ξ ∈ ℝ^{N} is assumed, not only for |ξ| = γ. Lemma 2.2 will be applied in form of the next corollary.

Corollary 2.3. Let B: L_{∞,c} → L_{1,loc} be a linear operator. Assume that

||e^{ξx}Be^{−ξx}||_{p→q} ⩽ Mt^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q})}e^{μ|ξ|^{m}t} (ξ ∈ ℝ^{N})

for some 1 ⩽ p ⩽ q ⩽ ∞, M, t, μ > 0. Then

(a) ||B||_{r→r} ⩽ M_{1} := Mec_{N}μ^{\frac{N}{m} ( \frac{1}{p} − \frac{1}{q})} for all r ∈ [p, q].

(b) For all p ⩽ r ⩽ s ⩽ q we have

||e^{ξx}Be^{−ξx}||_{r→s} ⩽ M_{1}t^{− \frac{N}{m} ( \frac{1}{r} − \frac{1}{s})}e^{μ_{1}|ξ|^{m}t} (ξ ∈ ℝ^{N}),

with μ_{1} = 2^{m}μ.

Proof. (a) By Lemma 2.2 we have, choosing γ = (μt)^{−1/m}:

||B||_{r→r} ⩽ c_{N}(μt)^{\frac{N}{m} ( \frac{1}{p} − \frac{1}{q})} · Mt^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q})}e^{1}
= c_{N}μ^{\frac{N}{m} ( \frac{1}{p} − \frac{1}{q})}Me.

(b) Let ξ ∈ ℝ^{N}. For B_{ξ} := e^{ξx}Be^{−ξx} and ξ_{0} ∈ ℝ^{N} we have by assumption that

||e^{ξ_{0}x}B_{ξ}e^{−ξ_{0}x}||_{p→q} ⩽ Mt^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q})}e^{μ|ξ+ξ_{0}|^{m}t}.

By (a) we conclude, noting |ξ + ξ_{0}|^{m} ⩽ 2^{m}(|ξ|^{m} + |ξ_{0}|^{m}),

||B_{ξ}||_{r→r} ⩽ M_{1}e^{μ2^{m}|ξ|^{m}t}.

Riesz-Thorin interpolation between this inequality and the assumption of the corollary leads to the desired conclusion.

Proposition 2.4. Let T be a C_{0}-semigroup on L_{q} and assume that

||e^{ξx}T (t)e^{−ξx}||_{p→q} ⩽ Mt^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q})}e^{μ|ξ|^{m}t} (t > 0, ξ ∈ ℝ^{N})

for some 1 ⩽ p ⩽ q, M, μ > 0. Then T extrapolates to a C_{0}-semigroup on L_{p}.

Proof. It follows from Corollary 2.3(a) that T extrapolates to a bounded semigroup on L_{p}. Thus, it suffices to show that T (t)f → f in L_{p} as t → 0, for all f ∈ L_{∞,c} with ||f||_{p} = 1. By Corollary 2.3(b) (with r = s = p) we have

||e^{ξx}T (t)f||_{p} ⩽ M_{1}e^{μ_{1}|ξ|^{m}t}||e^{ξx}f||_{p} (t ⩾ 0, ξ ∈ ℝ^{N}).

Let t ⩽ 1, |ξ| = 1. Then ||e^{ξx}T (t)f||_{p} ⩽ M_{1}e^{μ_{1}}||e^{|x|}f||_{p} =: c < ∞ since f has compact support. Let R > 0 and χ_{ξ} the characteristic function of the set {x ∈ Ω; ξx ⩾ R}. Then ||χ_{ξ}T (t)f||_{p} ⩽ ||e^{ξx−R}T (t)f||_{p} ⩽ ce^{−R}. Let K_{R} be the cube of edge length 2R centered at 0. Then, with e_{j} being the standard unit vectors of ℝ^{N},

||χ_{Ω\K_{R}}T (t)f||_{p} ⩽ || ∑_{j=1}^{N}(χ_{e_{j}} + χ_{−e_{j}})T (t)f||_{p} ⩽ 2Nce^{−R}.

For R so large that supp f ⊆ K_{R} it follows that

||T (t)f − f||_{p} ⩽ ||χ_{Ω∩K_{R}}T (t)f − f||_{p} + ||χ_{Ω\K_{R}}T (t)f||_{p}
⩽ |Ω ∩ K_{R}|^{\frac{1}{p} − \frac{1}{q}} ||T (t)f − f||_{q} + 2Nce^{−R},

which proves the assertion.

Remark. For p > 1 or in case T is positive, the above proposition follows directly from Corollary 2.3(a) and [28].

Until now we have used weighted estimates with weights of the form ρ(x) = e^{ξx}. Generally, we call ρ: Ω → (0, ∞) a weight function if ρ, ρ^{−1} ∈ L_{∞,loc}. In the proof of Theorem 2.1 we need to extend the weighted estimate (2.1) from real to complex times. The next proposition serves this purpose. Comparable results are shown in [4], [18] and [9] by means of the Phragmen-Lindelöf theorem on a sector. But it seems to be more natural to use the Stein interpolation on a strip, similar to the proof of [6, Lemma 9] by means of the three lines theorem.

Proposition 2.5. Let ρ: Ω → (0, ∞) be a weight function, θ ∈ (0, \frac{π}{2}], S_{θ} := {0 ≠ z ∈ ℂ; |arg z| < θ}. 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}|| ⩽ Me^{μγ^{m}t} (t > 0, γ ⩾ 0)

for some M ⩾ 1, μ > 0, m > 1. Then, for α ∈ (0, θ), there exists μ_{α} > 0 such that

||ρ^{γ}F (z)ρ^{−γ}|| ⩽ M_{1}e^{μ_{α}γ^{m} Re z} (z ∈ S_{α}, γ ⩾ 0),

with M_{1} = max{||F ||_{∞}, M}.

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

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

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 adapted to have ||G(z)|| ⩽ M_{1} = max{||F ||_{∞}, M} for Re z = 0, 1. We infer that ||G(z)|| ⩽ M_{1} for all 0 ⩽ Re z ⩽ 1, so

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

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

||ρ^{xγ}F (z)ρ^{−xγ}|| ⩽ M_{1} exp(μγ^{m} \frac{sin(θ−α)}{sin θ} \frac{Re z}{cos α}).

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

Proof of Theorem 2.1. Without restriction let ω = 0. Observe that for the first assertion it suffices to consider the case p = r ∧ r_{0}, q = r ∨ r_{0}, by Corollary 2.3(b). We confine ourselves to the case r < r_{0} (so that p = r, q = r_{0}), the proof of the case r > r_{0} being almost the same.

By Proposition 2.4, T (t)↾_{L_{∞,c}} extends to a C_{0}-semigroup on L_{p}. Let 0 < α < θ. Note that the function S_{α} ∋ z ↦ ⟨T (z)f, g⟩ is analytic for all f, g ∈ L_{∞,c} and that L_{∞,c} is dense in L_{p} and a norming subset of L_{p}^{∗}. So we only have to show that ||T (z)↾_{L_{∞,c}}||_{p→p} ⩽ M_{α} for |arg z| ⩽ α to conclude the assertion by a slight modification of [10, Thm. III.1.12].

From assumption (2.1) and Corollary 2.3(b) we obtain that

||e^{ξx}T (t)e^{−ξx}||_{q→q} ⩽ Ce^{μ_{1}|ξ|^{m}t} (t ⩾ 0, ξ ∈ ℝ^{N}).

Let α_{1} := \frac{α+θ}{2}, and δ > 0 be such that z − δ Re z ∈ S_{α_{1}} for all z ∈ S_{α}. For z ∈ S_{α}, ξ ∈ ℝ^{N} and f ∈ L_{∞,c} we obtain, taking into account Proposition 2.5 and assumption (2.1),

||e^{ξx}T (z)e^{−ξx}f||_{q} = ||e^{ξx}T (z − δ Re z)e^{−ξx}e^{ξx}T (δ Re z)e^{−ξx}f||_{q}
⩽ M_{1}e^{μ_{α_{1}}|ξ|^{m} Re(z−δ Re z)}M(δ Re z)^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q})}e^{μ|ξ|^{m}δ Re z}||f||_{p}
= M_{2}(Re z)^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q})}e^{μ_{2}|ξ|^{m} Re z}||f||_{p},

with μ_{2} = (1 − δ)μ_{α_{1}} + δμ. An application of Corollary 2.3(a) yields the first assertion.

The statement on p-independence of the spectra follows from [16, Sec. 5, 1.].

In applications of Theorem 2.1 it is often hard to verify the weighted estimate (2.1) for p = 1. The next result serves the purpose to overcome this difficulty.

Proposition 2.6. Let r_{0} ⩾ 1, T a contractive C_{0}-semigroup on L_{r_{0}}, and ρ > 0 a weight function. Assume that

||ρ^{γ}T (t)ρ^{−γ}||_{p→q} ⩽ Mt^{−α( \frac{1}{p} − \frac{1}{q})}e^{μγ^{m}t} (t, γ > 0)

for some r_{0} < p < q, M, α, μ > 0, m > 1. Then there exist M_{1}, μ_{1} > 0 such that

||ρ^{γ}T (t)ρ^{−γ}||_{r_{0}→q} ⩽ M_{1}t^{−α( \frac{1}{r_{0}} − \frac{1}{q})}e^{μ_{1}γ^{m}t} (t, γ > 0).

Proof. For 0 < θ ⩽ 1 let p_{θ} := (\frac{θ}{p} + \frac{1−θ}{r_{0}})^{−1}, q_{θ} := (\frac{θ}{q} + \frac{1−θ}{r_{0}})^{−1}. By the Stein interpolation theorem, the assumption implies that

||ρ^{θγ}T (t)ρ^{−θγ}||_{p_{θ}→q_{θ}} ⩽ M^{θ}t^{−θα( \frac{1}{p} − \frac{1}{q})}e^{θμγ^{m}t} (t, γ > 0).  (2.2)

Let t, γ > 0, define θ ∈ (0, 1) by 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 (2.2) yields

||ρ^{γ}T (t_{k})ρ^{−γ}||_{q_{θ_{k+1}}→q_{θ_{k}}} ⩽ M^{θ_{k}}t_{k}^{−θ_{k}β}e^{θ_{k}μ(γ/θ_{k})^{m}t_{k}} = M^{θ_{k}}(θ^{mk}t)^{−θ_{k}β}e^{θ_{k}μγ^{m}t}

for all k ∈ ℕ_{0}. We use this as a starting point for a Moser type iteration: for f ∈ L_{∞,c} 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}

⩽ lim inf_{n→∞} ∏_{k=0}^{n}(M^{θ_{k}}θ^{−mβkθ_{k}}t^{−θ_{k}β}e^{θ_{k}μγ^{m}t}) · ||f||_{q_{θ_{n+1}}}.

Set r := ∑_{k=0}^{∞} θ_{k}(= \frac{1}{1−θ}) and s := ∑_{k=0}^{∞} kθ_{k}(= \frac{θ}{(1−θ)^{2}} ), and note that ∑_{k=0}^{∞} θ_{k}β = α(\frac{1}{r_{0}} − \frac{1}{q}). We conclude that

||ρ^{γ}T (\frac{t}{1−θ^{m}})ρ^{−γ}f||_{q} ⩽ M^{r}θ^{−mβs}t^{−α( \frac{1}{r_{0}} − \frac{1}{q})}e^{rμγ^{m}t}||f||_{r_{0}}.

This yields the assertion with M_{1} = M^{r}θ^{−mβs}(1 − θ^{m})^{−α( \frac{1}{r_{0}} − \frac{1}{q})} and μ_{1} = (1 − θ^{m})rμ.

The next extrapolation lemma is a modification of the result from [3] with literally the same proof.

Lemma 2.7. Let p_{0} ⩽ p < q ⩽ p_{1}. Let T be a semigroup 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).

Then there exists C_{1} > 0 such that

||T (t)||_{p_{0}→p_{1}} ⩽ C_{1}t^{−α( \frac{1}{p_{1}} − \frac{1}{p_{0}} )} (t > 0).

In the next section we apply Theorem 2.1 via the following proposition.

Proposition 2.8. Let 1 ⩽ p_{0} < ∞, T an analytic semigroup of angle θ ∈ (0, \frac{π}{2}] on L_{p_{0}} satisfying

||e^{ξx}T (t)e^{−ξx}||_{p_{0}→p_{0}} ⩽ Me^{μ|ξ|^{m}t+ωt},
||T (t)||_{p→q} ⩽ Mt^{− \frac{N}{m} ( \frac{1}{p} − \frac{1}{q})}e^{ωt},

for all t > 0, ξ ∈ ℝ^{N} and some 1 ⩽ p < q ⩽ ∞. Then T (t)↾_{L_{∞,c}} extends to an analytic semigroup of angle θ on L_{r} for r ∈ (p∧p_{0}, q∨p_{0})∪{p_{0}}, and the spectrum of the generators −A_{r} is independent of r. If in addition T is L_{r_{0}}-contractive for some 1 ⩽ r_{0} < p_{0}, the same holds for r ∈ [r_{0}, p_{0}].

Proof. Denote ρ(x) = e^{ξx}. Without restriction let ω = 0. By the Stein interpolation theorem the assumptions imply that, for all θ ∈ (0, 1),

||e^{ξx}T (t)e^{−ξx}||_{p_{θ}→q_{θ}} ⩽ Mt^{− \frac{N}{m} ( \frac{1}{p_{θ}} − \frac{1}{q_{θ}})}e^{θ^{1−m}μ|ξ|^{m}t},

with \frac{1}{p_{θ}} = \frac{1−θ}{p} + \frac{θ}{p_{0}} and \frac{1}{q_{θ}} = \frac{1−θ}{q} + \frac{s}{p_{0}} . In the rest of the proof we distinguish between three cases.

Case 1. p ⩽ p_{0} ⩽ q. The assertion follows directly from Theorem 2.1.

Case 2. p_{0} < p. By Corollary 2.3(b) we have that

||e^{ξx}T (t)e^{−ξx}||_{q_{θ}→q_{θ}} ⩽ Me^{μ_{θ}|ξ|^{m}t}.

Then Lemma 2.7 (applied to the semigroup T_{ξ}(t) = e^{ξx}T (t)e^{−ξx}) and Theorem 2.1 yield the assertion.

Case 3. p_{0} > q. The proof is analogous to that of Case 2.

The last assertion is obtained in the same way, using Proposition 2.6.

3 Proof of the main result

In this section we prove Theorem 1.2. In order to apply Proposition 2.8, we need to show appropriate weighted estimates for the semigroups T_{p} constructed in Theorem 1.1. Recall that the semigroups T_{p} are associated with the form τ defined in (1.1). We will establish estimates on the ‘twisted semigroups’ e^{ξx}T_{p}e^{−ξx}, for ξ ∈ ℝ^{N}, by studying the ‘twisted form’ τ_{ξ} which is formally defined by τ_{ξ}(u, v) = τ(e^{−ξx}u, e^{ξx}v). We point out that it is a nontrivial technical problem to establish the relationship between τ_{ξ} and e^{ξx}T_{p}e^{−ξx} (see, e.g., [20, Prop. 3.4]).

Throughout this section we assume that (a), (bV) and (BC) are fulfilled and that a ∈ L_{∞}. Let τ_{a}, τ, τ_{p} (1 ⩽ p < ∞) be the forms defined in Section 1. Recall that

I = {p ∈ [1, ∞); τ_{p} ⩾ −ω_{p} for some ω_{p} ∈ ℝ}.

For a Lipschitz continuous function ϕ: Ω → ℝ, we introduce the form

τ_{ϕ}(u, v) := τ(u, v) − ⟨(a∇ϕ)u, ∇v⟩ + ⟨∇u, (a^{⊤}∇ϕ)v⟩
− ⟨[a_{s}∇ϕ · ∇ϕ + (b_{1} + b_{2})∇ϕ]u, v⟩

on D(τ_{ϕ}) := D(τ). It is straightforward that

τ_{ϕ}(u, v) = τ(e^{−ϕ}u, e^{ϕ}v) (u, v ∈ D(τ_{ϕ}) such that e^{−ϕ}u, e^{ϕ}v ∈ D(τ)).

The form τ_{ϕ} is of the same type as the form τ, with new lower order coefficients

b̃_{1} = b_{1} + a^{⊤}∇ϕ, b̃_{2} = b_{2} + a∇ϕ, Ṽ = V − a_{s}∇ϕ · ∇ϕ − (b_{1} + b_{2})∇ϕ.

Since a ∈ L_{∞} and ∇ϕ ∈ L_{∞}, it is easy to see that these new coefficients satisfy assumption (bV).

Proposition 3.1. Assume that (a), (bV) and (BC) hold, and recall that a ∈ L_{∞}. Let c > 0, 0 < ε < \frac{1}{2}, p ∈ ^{∘} I, and T_{p} the positive C_{0}-semigroup on L_{p} associated with τ. Then there exists μ > 0 such that, for all Lipschitz continuous ϕ: Ω → ℝ satisfying

|⟨(b_{1} + b_{2})∇ϕ, u^{2}⟩| ⩽ ετ_{p}(u) + c(1 + ||∇ϕ||_{∞}^{2})||u||_{2}^{2} (0 ⩽ u ∈ D(τ)),  (3.1)

the following assertions hold.

The form τ_{ϕ} is associated with a positive C_{0}-semigroup T_{ϕ,p}(t) = e^{−tA_{ϕ,p}} on L_{p}. For all u ∈ D(A_{ϕ,p}) we have |u|^{\frac{p}{2}} sgn u ∈ D(\overline{τ_{p}}) and

⟨A_{ϕ,p}u, |u|^{p−1} sgn u⟩ ⩾ (1 − 2ε)\overline{τ_{p}}(|u|^{\frac{p}{2}} sgn u) − μ(1 + ||∇ϕ||_{∞}^{2})||u||_{p}^{p}.

Further, T_{ϕ,p}(t)f = e^{ϕ}T_{p}(t)e^{−ϕ}f for all f ∈ L_{∞,c}, t ⩾ 0. In particular,

||e^{ϕ}T_{p}(t)e^{−ϕ}||_{p→p} ⩽ e^{μ(1+||∇ϕ||_{∞}^{2})t} (t ⩾ 0).

For the proof of the proposition, we need the following technical lemma.

Lemma 3.2. Let ϕ ∈ L_{∞,loc}. Let τ, τ_{ϕ} be closed sectorial forms in L_{2} with D(τ) = D(τ_{ϕ}), and A, A_{ϕ} the corresponding m-sectorial operators in L_{2}. Assume that

τ_{ϕ}(u, v) = τ(e^{−ϕ}u, e^{ϕ}v) (u, v ∈ D(τ_{ϕ}) such that e^{−ϕ}u, e^{ϕ}v ∈ D(τ)).

Let λ ∈ ρ(−A_{ϕ}) ∩ ρ(−A). Then

(λ + A_{ϕ})^{−1}f = e^{ϕ}(λ + A)^{−1}e^{−ϕ}f (f ∈ L_{∞,c})

if and only if

e^{ϕ}v ∈ D(τ) (v ∈ (λ + A)^{−1}L_{∞,c}).

Proof. The “only if” part is clear, so we prove the “if” part. Let f ∈ L_{∞,c}, u := e^{ϕ}(λ + A)^{−1}e^{−ϕ}f. Then e^{−ϕ}u ∈ (λ + A)^{−1}L_{∞,c} =: D ⊆ D(τ) and hence u ∈ D(τ). For all v ∈ D we have e^{ϕ}v ∈ D(τ), so

τ_{ϕ}(u, v) = τ(e^{−ϕ}u, e^{ϕ}v) = ⟨e^{ϕ}Ae^{−ϕ}u, v⟩.

Moreover, since D(τ) = D(τ_{ϕ}), the closed graph theorem implies that D is a core for τ_{ϕ}. Thus we obtain that u ∈ D(A_{ϕ}) and A_{ϕ}u = e^{ϕ}Ae^{−ϕ}u, which implies the assertion.

Proof of Proposition 3.1. In order to apply Theorem 1.1 we have to consider the symmetric form τ_{ϕ,p} defined by

τ_{ϕ,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]|u|, ∇|u|⟩ − ⟨[a_{s}∇ϕ ·∇ϕ + (b_{1} + b_{2}) ·∇ϕ]|u|^{2}⟩

on D(τ_{ϕ,p}) := D(τ_{ϕ}). By assumption (a) we have |⟨aζ, η⟩| ⩽ (α+1)|a_{s}^{1/2}ζ|·|a_{s}^{1/2}η| for all ζ, η ∈ ℂ^{N}. A standard quadratic estimate shows that

τ_{ϕ,p}(u) ⩾ τ_{p}(u) − δ(α + 1)^{2}τ_{a}(|u|) − ⟨[(1 + \frac{1}{δ})a_{s}∇ϕ · ∇ϕ + (b_{1} + b_{2})∇ϕ]|u|^{2}⟩

for all δ > 0, u ∈ D(τ). By [24, Prop. 4.1(b)] there exist δ > 0, ω ∈ ℝ such that

δ(α + 1)^{2} Re τ_{a} ⩽ ετ_{p} + ω.

By (3.1) we thus obtain

τ_{ϕ,p} ⩾ (1 − 2ε)τ_{p} − ω − (1 + \frac{1}{δ})||a_{s}||_{∞}||∇ϕ||_{∞}^{2} − c(1 + ||∇ϕ||_{∞}^{2}).

An application of Theorem 1.1 completes the proof of the first two assertions.

Let now U_{ϕ} := (α + 1)^{2}||a_{s}||_{∞}||∇ϕ||_{∞}^{2} + W_{1} + W_{2} + |V |. Then U := 5U_{ϕ} is τ_{a}-regular by assumption (bV). Standard quadratic estimates show that

Re τ_{ϕ} ⩾ \frac{1}{4} Re τ_{a} − 4U_{ϕ}  (3.2)

and that τ + U, τ_{ϕ} + U are densely defined closed sectorial forms, with domains D(τ_{a} + U_{ϕ}). For m ∈ ℕ let U_{m} := (U − m)^{+}, and A_{m}, A_{ϕ,m} the m-sectorial operators associated with τ + U_{m}, τ_{ϕ} + U_{m}, respectively. Due to Theorem 1.1(ii), the last assertion of the proposition will follow by passing to the limit in

e^{−tA_{ϕ,m}}f = e^{ϕ}e^{−tA_{m}}e^{−ϕ}f (f ∈ L_{∞,c}, t ⩾ 0).

This in turn is equivalent to

(λ + A_{ϕ,m})^{−1}f = e^{ϕ}(λ + A_{m})^{−1}e^{−ϕ}f (m ∈ ℕ, λ > m, f ∈ L_{∞,c}).

Thus, by Lemma 3.2, it remains to show that

e^{ϕ}v ∈ Q := D(τ_{a} + U_{ϕ}) for all v ∈ D := (λ + A_{m})^{−1}L_{∞,c}.  (3.3)

For n ∈ ℕ let ϕ_{n} := ϕ ∧ n. It is easy to see that τ_{ϕ_{n}} + U_{m} is a densely defined closed sectorial form with domain Q. Let A_{ϕ_{n},m} denote the m-sectorial operator associated with τ_{ϕ_{n}} + U_{m}. By (3.2) we estimate

Re τ_{ϕ_{n}} + U_{m} ⩾ \frac{1}{4} Re τ_{a} − 4U_{ϕ_{n}} + 5U_{ϕ} − m ⩾ \frac{1}{4} Re τ_{a} + U_{ϕ} − m ⩾ −m

Let g ∈ L_{∞,c}, v := (λ + A_{m})^{−1}g. Note that ϕ_{n} ∈ W_{∞}^{1}. Hence, by assumption (BC), we conclude from Lemma 3.2 that

(λ + A_{ϕ_{n},m})^{−1}(e^{ϕ}g) = e^{ϕ_{n}}(λ + A_{m})^{−1}e^{−ϕ_{n}}(e^{ϕ_{n}}g) = e^{ϕ_{n}}v

for all λ > m and sufficiently large n ∈ ℕ. Therefore,

(\frac{1}{4} Re τ_{a} + U_{ϕ})(e^{ϕ_{n}}v) ⩽ Re(τ_{ϕ_{n}} + U_{m} + λ)(e^{ϕ_{n}}v) = Re⟨e^{ϕ}g, e^{ϕ_{n}}v⟩ ⩽ \frac{1}{λ−m}||e^{ϕ}g||_{2}^{2}.

This shows that (e^{ϕ_{n}}v) is a bounded sequence in Q. Moreover, (|e^{ϕ_{n}}v|) is pointwise increasing, and e^{ϕ_{n}}v → e^{ϕ}v a.e. as n → ∞. Hence e^{ϕ}v ∈ L_{2} by monotone convergence, and e^{ϕ_{n}}v → e^{ϕ}v in L_{2} by dominated convergence. We conclude that e^{ϕ}v ∈ Q, i.e., (3.3) holds.

Proof of Theorem 1.2. Let p ∈ (p_{−}, p_{+}), T_{p}(t) = e^{−A_{p}t} be the semigroup on L_{p} associated with the form τ. For ξ ∈ ℝ^{N} let ϕ_{ξ}(x) := ξ · x. Then ∇ϕ_{ξ} = ξ. By assumption (ii) of the theorem and Euclid’s inequality, we have

|⟨(b_{1} + b_{2})ξ, u^{2}⟩| ⩽ |ξ| · |⟨(b_{1} + b_{2})u^{2}⟩| ⩽ \frac{1}{4}(τ_{p} + C)(u) + C^{2}|ξ|^{2}||u||_{2}^{2}.

So we can apply Proposition 3.1 to the form τ_{ϕ_{ξ}} and obtain that

||e^{ξx}T_{p}(t)e^{−ξx}||_{p→p} ⩽ e^{μ(1+|ξ|^{2})t} (t ⩾ 0, ξ ∈ ℝ^{N}),

which verifies the first assumption of Proposition 2.8. Now we are going to establish an estimate on ||T_{p}(t)||_{p→ \frac{N}{N−2} p}.

By [24, Prop. 4.1(b)], there exist ε_{p} > 0 and C_{p} ∈ ℝ such that

τ_{p} ⩾ ε_{p} Re τ_{a} + C_{p}.  (3.4)

Without restriction C_{p} = 1. Let 0 ⩽ f ∈ L_{p}, t ⩾ 0, u := e^{−A_{p}t}f. Then 0 ⩽ u ∈ D(A_{p}) since T_{p} is positive and analytic.

By Theorem 1.1(iii), (3.4), and assumption (iii) of the theorem, there exists δ > 0 such that

⟨A_{p}u, u^{p−1}⟩ ⩾ ε_{p}τ_{a}(u^{p/2}) + ||u^{p/2}||_{2}^{2} ⩾ δ||u||_{\frac{N}{N−2} p}^{p}.

Using the analyticity of T_{p} we obtain by Hölder’s inequality that

⟨A_{p}u, u^{p−1}⟩ ⩽ \frac{C}{t} ||f||_{p}^{p},

with some C > 0 not depending on t. Combining the above two estimates we arrive at ||u||_{\frac{N}{N−2} p} ⩽ C_{1}t^{− \frac{1}{p}} ||f||_{p}, so that

||T (t)||_{p→ \frac{N}{N−2} p} ⩽ C_{1}t^{− \frac{1}{p}} .  (3.5)

Applying now Proposition 2.8 (note that \frac{1}{p} = \frac{N}{2} (\frac{1}{p} − \frac{N−2}{Np})), we infer the assertion of the theorem for p ∈ (p_{−}, p_{max}) (and, in case 1 ∈ I, for p ∈ [1, p_{max})).

The p→q estimate (1.5) follows from (3.5) and Lemma 2.7.

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}). In order to complete the proof in the case 1 ∉ I, one should repeat the arguments for the adjoint semigroup T^{∗} which is associated with the form τ^{∗} (see [24, Prop. 3.11]).

4 Sharpness of the main theorem

In this section we give an example of a semigroup for which the interval in the L_{p}-scale obtained in Corollary 1.3 cannot be extended.

Let b: Ω → ℝ^{N}, V: Ω → ℝ be such that H_{0}^{1} ∩ Q(|b|^{2} + |V |) is dense in H_{0}^{1}. Define the form τ in L_{2} by

τ(u, v) = ⟨∇u, ∇v⟩ + ⟨b∇u, v⟩ + ⟨V u, v⟩

on D(τ) := H_{0}^{1} ∩ Q(|b|^{2} + |V |).

Proposition 4.1. Assume that τ is associated with a C_{0}-semigroup e^{−A_{p}t} on L_{p} for some p ⩾ 1. Then

D(A_{p}) ⊇ D_{p} := {u ∈ H_{0}^{2} ∩ W_{p}^{2}; |b| |∇u|, |b|^{2}u, V u ∈ L_{2} ∩ L_{p}}

and A_{p} ⊇ (−Δ + b∇ + V)↾_{D_{p}}.

Proof. Define the operator L by Lu = (−Δ + b∇ + V)u, D(L) = D_{p}. Then L acts in both L_{2} and L_{p}. Let U_{0} := |b|^{2} + 2|V |, and let e^{−A_{0}t} be the semigroup on L_{2} associated with the closed sectorial form τ + U_{0}. Then e^{−A_{0}t} extrapolates to a C_{0}-semigroup e^{−A_{0,p}t} on L_{p}.

It is easy to see that D_{p} ⊆ D(τ + U_{0}) and

(τ + U_{0})(u, v) = ⟨(L + U_{0})u, v⟩ (u ∈ D_{p}, v ∈ D(τ)).

Hence A_{0} ⊇ L + U_{0} and, moreover, A_{0,p} ⊇ L + U_{0} since L + U_{0} is an operator in L_{p}. By [27, Cor. 2.7] we conclude that A_{p} ⊇ A_{0,p} − U_{0} ⊇ L.

In the following we denote r(x) := |x|.

Corollary 4.2. Let Ω = ℝ^{N}, b = c_{1}r^{−1}∇r, V = c_{2}r^{−2} + r^{2} and u = r^{−σ}e^{− \frac{r^{2}}{2}} , σ ∈ ℝ. Assume that τ is associated with a C_{0}-semigroup e^{−A_{p}t} on L_{p}, for some p ∈ [1, ∞) satisfying p(σ + 2) < N. Then u ∈ D(A_{p}) and

A_{p}u = (−(σ^{2} − (N − 2 − c_{1})σ − c_{2})r^{−2} + N − c_{1} − 2σ)u.

Proof. Note that Δu, \frac{∇u}{r}, \frac{u}{r^{2}} and r^{2}u belong to L_{1} ∩ L_{∞}(ℝ^{N} \ B_{ε}) for all ε > 0, where B_{ε} = {x ∈ ℝ^{N}; |x| < ε}. Let φ ∈ C^{∞}(ℝ^{N}), 0 ⩽ φ ⩽ 1, φ(x) = 1 for all x ∈ B_{1}^{c}, φ(x) = 0 for all x ∈ B_{1/2}. Let φ_{n}(x) := φ(nx), u_{n} := φ_{n}u. Then, by Proposition 4.1, u_{n} ∈ D(A_{p}) and

A_{p}u_{n} = (−Δ + b∇ + V)u_{n}
= φ_{n}(−Δ + b∇ + V)u − 2∇φ_{n} · ∇u + (b · ∇φ_{n} − Δφ_{n})u.

Since supp(1 − φ_{n}) ⊆ B_{\frac{1}{n}}, we have |Δφ_{n}(x)| ⩽ |Δφ|(nx)r^{−2} and |∇φ_{n}(x)| ⩽ |∇φ|(nx)r^{−1}. Moreover, Δu, \frac{∇u}{r}, \frac{u}{r^{2}} ∈ L_{p} since σ + 2 < \frac{N}{p}. Hence A_{p}u_{n} → (−Δ + b∇ + V)u in L_{p}, by the dominated convergence theorem. So u ∈ D(A_{p}) and A_{p}u = (−Δ + b∇ + V)u since u_{n} → u in L_{p} and A_{p} is a closed operator. The second assertion now results from a direct computation.

Let now b = β \frac{N−2}{2}r^{−1}∇r and V = −γ \frac{(N−2)^{2}}{4} r^{−2} + r^{2} with β < 2, 0 < γ < (1 − β/2)^{2}. Let μ := \sqrt{(1 − β/2)^{2} − γ}. Then by Corollary 1.3, τ is associated with a consistent family of C_{0}-semigroups e^{−A_{p}t} on L_{p}, for all

p_{min} := \frac{2N}{4 + (N − 2)(1 − \frac{β}{2} + μ)} < p < \frac{2N}{(N − 2)(1 − \frac{β}{2} − μ)} =: p_{max}.

We are going to show that, for q ∉ (p_{min}, p_{max}), the semigroup e^{−A_{p}t} does not extrapolate to a C_{0}-semigroup on L_{q}. Let

σ := \frac{N}{p_{max}} = \frac{N − 2}{2}(1 − \frac{β}{2} − μ), p_{0} := \frac{N}{σ + 2} = \frac{2N}{4 + (N − 2)(1 − \frac{β}{2} − μ)}.

Then p_{0} ∈ (p_{min}, p_{max}). By Corollary 4.2, u = r^{−σ}e^{− \frac{r^{2}}{2}} is an eigenfunction of A_{p} for p ∈ (p_{min}, p_{0}). Now assume that e^{−A_{p}t} extrapolates to a semigroup on L_{q}, for some q ⩾ p_{max}. Then, by (1.5) and Lemma 2.7, e^{−A_{p}t}: L_{p} → L_{q} for all p ∈ (p_{min}, p_{max}). In particular, e^{−A_{p}t}u ∈ L_{q}. This contradicts the fact that e^{−A_{p}t}u = e^{ct}u ∉ L_{q} (recall σ = \frac{N}{p_{max}} ⩾ \frac{N}{q}). Considering the adjoint semigroup we show that e^{−A_{p}t} does not extrapolate to a semigroup on L_{q}, for any q ⩽ p_{min}.

Remark. In the case of Schrödinger semigroups, a similar example was given by Yu. Semenov (private communication).

5 Non-divergence type operators

In this section we consider the operator

A = −a∇^{2} = − ∑_{j,k=1}^{N} a_{jk} \frac{∂^{2}}{∂x_{j}∂x_{k}}

in UC_{b}(ℝ^{N}), the space of bounded uniformly continuous functions, with D(A) = UC_{b}^{2}(ℝ^{N}) (the functions and their first and second derivatives are in UC_{b}(ℝ^{N})). We assume that (a_{jk}) is symmetric with smooth entries and that σ^{−1} id ⩽ a ⩽ σ id for some σ ⩾ 1. It is well-known that the closure of −A generates an analytic semigroup T of full angle (i.e., of angle \frac{π}{2}) on UC_{b}(ℝ^{N}) (see, e.g., [13, Thm. 8.2.1]).

The semigroup operators T (t) are integral operators with smooth integral kernels p(t) satisfying

∫_{ℝ^{N}} p(t, x, y)dy = 1.

The adjoint semigroup T^{∗} on L_{1} is defined by

(T^{∗}(t)f)(y) = ∫_{ℝ^{N}} p(t, x, y)f(x)dx.

It was proved in [7] that there exist q = q(σ, N) > \frac{N}{N−1} and C = C(σ, N) such that

sup_{x∈ℝ^{N}} ||p(t, x, ·)||_{q} < Ct^{− \frac{N}{2q′}} ,

which implies that

||T^{∗}(t)||_{1→q} ⩽ Ct^{− \frac{N}{2q′}} .  (5.1)

Now we introduce the ‘weighted semigroups’. Let ξ ∈ ℝ^{N}, ρ_{ξ}(x) := e^{ξx}. Then

(ρ_{ξ}^{−1}T (t)ρ_{ξ}f)(x) = ∫_{ℝ^{N}} e^{−ξx}p(t, x, y)e^{ξy}f(y)dy

Using the maximum principle we see that (cf. [25])

||ρ_{ξ}^{−1}T (t)ρ_{ξ}||_{∞→∞} ⩽ e^{σ|ξ|^{2}t} so that ||ρ_{ξ}T^{∗}(t)ρ_{ξ}^{−1}||_{1→1} ⩽ e^{σ|ξ|^{2}t}.  (5.2)

Estimates (5.1) and (5.2) allow us to apply Proposition 2.8.

Proposition 5.1. Let α ∈ [0, \frac{π}{2}). There exists a constant C_{α} depending only on α, N, σ such that

||e^{−\overline{A}z}||_{p→p} ⩽ C_{α} (p ∈ [N, ∞), |arg z| ⩽ α).

In particular, the family e^{−\overline{A}t}↾_{C_{c}} extends to an analytic semigroup of full angle on L_{p}.

Remark. For α = 0, the above proposition was first proved in [22].

The estimate obtained in Proposition 5.1 is an a-priori estimate which carries over to semigroups associated with the non-divergence form operator A that are obtained by approximation by semigroups corresponding to operators with smooth coefficients. We stress, however, that the above result does not contribute to the problem of solvability of non-divergence type equations for non-smooth a.

At the same time, the main results of this paper can be applied to the problem of well-posedness of the abstract Cauchy problem in L_{p}(ℝ^{N}) for an operator realization corresponding to the non-divergence type elliptic differential expression A = −a∇^{2}.

Assume that (a_{jk}) is uniformly elliptic. Set b_{1,k} = ∑_{j=1}^{N} ∂_{j}a_{jk} (k = 1, …, N). Suppose that b_{1} ∈ L_{1,loc},

||b_{1}u|| ⩽ K||u||_{H^{1}} for some K > 0, b_{1} = b_{11} + b_{12},
⟨|b_{11}|^{2}|u|^{2}⟩ ⩽ β||a_{s}^{1/2}∇u||_{2}^{2} + C_{β}||u||_{2}^{2} for some β ∈ [0, 4), C_{β} ⩾ 0,
div b_{12} ∈ L_{1,loc}(ℝ^{N}), (div b_{12})^{−} ∈ L_{∞}(ℝ^{N}).

Then A = −∇(a∇)+b_{1}∇. By Corollary 1.3 one can associate with A an analytic C_{0}-semigroup T_{p} on L_{p}(ℝ^{N}), for all p ∈ (\frac{2N}{2N−\sqrt{β}(N−2)}, ∞), with sector of analyticity independent of p. This result is a generalization of the corresponding result in [15] (for the case of a uniformly elliptic matrix (a_{jk})) in several directions: firstly, the interval of solvability in the L_{p}-scale is extended (and in fact is sharp, see Section 4); secondly, the conditions on b_{1} are relaxed; and thirdly, as follows from Corollary 1.3, the sector of analyticity is p-independent.

6 Remark on higher order operators

In this short section we show that, employing Theorem 2.1, one can obtain a result similar to Theorem 1.2 for higher order (non-symmetric) operators from the class of superelliptic operators studied by E. B. Davies [6]. We sketch the construction of these operators below and refer the reader to [6] for details.

Let m < \frac{N}{2} , H^{m} := W_{2}^{m}(ℝ^{N}). Let τ, with D(τ) = H^{m}, be a closed sectorial form in L_{2} which satisfies the Gårding inequality

\frac{1}{2}||(−Δ)^{m/2}f||_{2}^{2} ⩽ Re τ(f) ⩽ c||(−Δ)^{m/2}f||_{2}^{2} + c||f||_{2}^{2},  (6.1)

for some c > 0 and all f ∈ H^{m}. Let E_{m} denote the set of all bounded real-valued C^{∞}-functions ϕ on ℝ^{N} such that ||D^{α}ϕ||_{∞} ⩽ 1 for all α such that 1 ⩽ |α| ⩽ m. Given λ ∈ ℝ and ϕ ∈ E_{m}, let

τ_{λϕ}(f, g) = τ(e^{−λϕ}f, e^{λϕ}g) (f, g ∈ H^{m}).

We assume that

|τ_{λϕ}(f) − Re τ(f)| ⩽ \frac{1}{4} Re τ(f) + k(1 + |λ|^{2m})||f||_{2}^{2} (f ∈ H^{m}),  (6.2)

for some k > 0 independent of λ and ϕ.

Proposition 6.1. Let assumptions (6.1) and (6.2) hold. Then the analytic C_{0}- semigroup T (t) = e^{−At} on L_{2}, associated with τ, extrapolates to an analytic semigroup T_{p}(t) = e^{−A_{p}t} on L_{p}, for all \frac{2N}{N+2m} ⩽ p ⩽ \frac{2N}{N−2m}. The sector of analyticity of T_{p} and the spectrum σ(A_{p}) are p-independent.

Sketch of the proof. In order to apply Theorem 2.1 one needs to verify the estimate

||e^{λϕ}T (t)e^{−λϕ}||_{2→ \frac{2N}{N−2m}} ⩽ \frac{c}{\sqrt{t}} e^{μ(|λ|^{2m}+1)t} (t > 0, λ ∈ ℝ, ϕ ∈ E_{m}),  (6.3)

for some c, μ > 0 (see [6, Lemma 4]). It follows from (6.1) and (6.2) that τ_{λϕ} is a closed sectorial form in L_{2}. By Lemma 3.2, the semigroup e^{λϕ}T (t)e^{−λϕ} is associated with τ_{λϕ}. Now a simple modification of the arguments in [6, Lemmata 6, 7, 22] leads to estimate (6.3).

Acknowledgment

Parts of this paper were written while the third author was visiting the School of Mathematics, University of Bristol. It is a pleasure to thank the department for the hospitality. Support of EPSRC through Grant GR/L76891 and of the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

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