Als Manuskript gedruckt

Technische Universität Dresden

Herausgeber: Der Rektor

Weighted norm estimates and L_{p}-spectral independence of linear

operators

Peer C. Kunstmann^{1} and Hendrik Vogt^{2}

2Institut für Analysis MATH-AN-02-00

Weighted norm estimates and L_{p}-spectral independence of linear operators

Peer C. Kunstmann^{1} and Hendrik Vogt^{2}

Abstract. We investigate the L_{p}-spectrum of linear operators defined consistently on L_{p}(Ω) for p_{0} ≤ p ≤ p_{1}, where (Ω, μ) is an arbitrary σ-finite measure space and 1 ≤ p_{0} < p_{1} ≤ ∞. We prove p-independence of the L_{p}-spectrum assuming weighted norm estimates or kernel estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω, μ) which is required to satisfy certain bounds for the volume growth of balls. Roughly speaking, these bounds are of subexponential type but we can also treat certain situations where the volume growth is exponential.

We show how previous results on L_{p}-spectral independence can be treated as special cases of our results and give examples – including strictly elliptic operators in Euclidean space, generators of semigroups that satisfy (generalized) Gaussian bounds, integral and non-integral regularizing operators – to indicate improvements.

Moreover, we study L_{p}-spectral independence in weighted L_{p}-spaces where the condition on the weight depends on the semi-metric d and is again of subexponential type.

AMS subject classification: 45P05, 35P05, 47A10, 47D03.

keywords: L^{p}-spectrum, weighted norm estimates, integral operators, resolvent, elliptic operators, heat kernel estimates.

1 Introduction and Main Results

Let (Ω, μ) be a σ-finite measure space and 1 ≤ p < q ≤ ∞. Suppose that A is a linear operator which acts consistently in L_{r}(Ω) for r ∈ [p, q], i.e.

A: L_{p}(Ω) ∩ L_{q}(Ω) → L_{p}(Ω) ∩ L_{q}(Ω)

is linear and extends, for all r ∈ [p, q], to a bounded operator A_{r}: L_{r}(Ω) → L_{r}(Ω) which, in addition, is weak^{∗}-continuous if r = ∞. The term “consistency” refers to the property

A_{r}f = A_{s}f (f ∈ L_{r}(Ω) ∩ L_{s}(Ω), r, s ∈ [p, q]).

It is then a natural question whether the spectrum σ(A_{r}) of A_{r} in L_{r}(Ω) depends on r ∈ [p, q] or not. This question also makes sense for unbounded operators A_{r} in L_{r}(Ω), r ∈ [p, q], if consistency is rephrased in terms of resolvents or semigroup operators (for the case that the A_{r} are generators). We refer to Remark 3(vi) for details.

There are several kinds of assumptions that are sufficient for r-independence of σ(A_{r}). We refer to the discussion in [14] and the references there. One method that has been

Typeset by L^{A}TEX 2ε

2supported by the Deutsche Forschungsgemeinschaft Address: ^{1}Mathematisches Institut I, Universität Karlsruhe, Englerstraße 2, 76128 Karlsruhe, Germany 2Technische Universität Dresden, Fachrichtung Mathematik, 01062 Dresden, Germany e-mail: ^{1}peer.kunstmann@math.uni-karlsruhe.de, ^{2}vogt@math.tu-dresden.de

Weighted norm estimates and L_{p}-spectral independence of linear operators 3

widely used in the context of the L_{p}-theory of elliptic operators relies on the exploitation of certain bounds, especially of Gaussian type, that hold for the kernels of associated integral operators, i.e. semigroup or resolvent operators ([9], [1], [20], [7], [10], [13], [14]). If these operators are not integral operators, which is in general the case if p > 1 and q < ∞, then r-independence of the spectrum still holds under the assumption of certain weighted norm estimates ([17], [16], [18], [15]). It is this kind of assumptions, i.e. weighted norm estimates and kernel bounds, that we shall use in this paper.

The case studied in most of the cited papers is that Ω is an open subset of ℝ^{N} and μ is the Lebesgue measure. We shall deal here with an arbitrary σ-finite measure space (Ω, μ). Our assumptions on the operator A, i.e. weighted norm estimates or kernel bounds, will be formulated in terms of a measurable semi-metric d on Ω which has suitable properties. The underlying philosophy is that, in order to get r-independence of the spectrum for a given operator A on (Ω, μ) by one of our results, one has to find a semi-metric d on Ω such that A and d satisfy our assumptions. With respect to the problem of r-independence of the spectrum as stated above this seems to be the natural procedure. Even if the space Ω carries a natural metric d_{0}, as open subsets of ℝ^{N} do or Riemannian manifolds ([20]), a suitable semi-metric d may be quite different. We want to emphasize that, in the case of weighted norm estimates, the consideration of a general semi-metric space (Ω, d) instead of (subsets of) ℝ^{N} means that we have to use a new approach in the proof which is rather different from the arguments used previously in this context.

In order to make the assumptions in our first main result (which uses weighted norm estimates) less restrictive we now slightly change the setting outlined above.

Let (Ω′, μ) be a σ-finite measure space and suppose that d is a measurable semi-metric on Ω′ satisfying

γ := ess inf_{z∈Ω′} μ(B(z, R)) > 0 for some R > 0,  (1)

where B(z, R) denotes the open ball with center z and radius R with respect to d. Further assume that

ess sup_{z∈Ω′} μ(B(z, r)) ≤ m(r) (r > 0),  (2)

where m: [0, ∞) → [0, ∞) is a non-decreasing function such that m(0) = 0 and for all ε > 0 there exists C_{ε} > 0 satisfying

m(r) ≤ C_{ε}e^{εr} (r > 0).  (3)

Note that we do not demand m(r) → 0 (r → 0), cf. [7, p. 177], and that (2) and (3) mean that the volume growth is uniformly subexponential. We point out that uniformly subexponential volume growth has a more restrictive definition in the context of Riemannian manifolds, see the end of Subsection 2.1.

Before we state our main theorems we introduce some notation. For a measurable function g on Ω′ × Ω′ and r ∈ [1, ∞] we define

n_{r}(g) := max ( ess sup_{x∈Ω′} ∥g(x, ·)∥_{L_{r}(Ω′)}, ess sup_{y∈Ω′} ∥g(·, y)∥_{L_{r}(Ω′)}) ∈ [0, ∞].

If n_{r}(g) is finite, g defines a bounded integral operator I_{g}: L_{p}(Ω′) → L_{q}(Ω′) for all 1 ≤ p ≤ q ≤ ∞ satisfying r^{−1} + p^{−1} = 1 + q^{−1}. In fact

∥I_{g}: L_{p}(Ω′) → L_{q}(Ω′)∥ ≤ n_{r}(g),  (4)

which can be proved by an application of Fubini’s theorem and Riesz-Thorin interpolation. In the case r = 1 the operator I_{g} is bounded in every space L_{p}(Ω′) (1 ≤ p ≤ ∞).

We define weight functions ρ_{ε,z} by

ρ_{ε,z}(x) := ρ_{ε}(z, x) := e^{−εd(z,x)} (x, z ∈ Ω′, ε ∈ ℝ).

Then n_{∞}(ρ_{ε}) ≤ 1 and by assumption (3)

n_{p}(ρ_{ε}) ≤ (∫_{0}^{∞} e^{−εpr} dm(r))^{1/p} < ∞ (1 ≤ p < ∞).  (5)

Here we have used the fact that, for any non-increasing function φ: [0, ∞) → [0, ∞) and almost all x ∈ Ω′, one can estimate

∫_{Ω′} φ(d(x, y)) dμ(y) ≤ ∫_{0}^{∞} φ(r) dm(r)

by assumption (2); cf. [7, p. 179].

Now let Ω ⊂ Ω′ be a measurable subset. We tacitly assume that functions defined on Ω are extended by 0 outside Ω when considered as functions on Ω′. In the following we consider operators on L_{p}(Ω), 1 ≤ p ≤ ∞. The reason for introducing the space Ω′ is that condition (1) is not assumed to hold for the balls of Ω but only for the (possibly larger) balls of Ω′. An important example for this situation is Ω′ = ℝ^{N} and a subset Ω ⊂ ℝ^{N}.

We fix an increasing sequence (Ω_{n}) of measurable subsets of Ω that have finite ddiameter and finite μ-volume (cf. Remark 3(viii)) such that Ω = ⋃_{n} Ω_{n}. By L_{1,loc}(Ω) we denote the set of (equivalence classes of) all measurable functions f on Ω with ∥f∥_{L_{1}(Ω_{n})} < ∞ for all n ∈ ℕ and by L_{∞,c}(Ω) the space of (equivalence classes of) all measurable functions f on Ω for which there is an n ∈ ℕ such that f = 0 a.e. on Ω \ Ω_{n} (see also Remark 3(v) for these spaces). Note that the weight functions ρ_{ε,z} are multiplicators on L_{1,loc}(Ω) and L_{∞,c}(Ω).

For a linear operator T: D(T) ⊂ L_{p}(Ω) → L_{q}(Ω), p, q ∈ [1, ∞], we denote its norm by ∥T ∥_{p→q}. The following are our main results. The first involves a weighted norm estimate.

Theorem 1. Assume that (1), (2) and (3) hold. Let 1 ≤ p < q ≤ ∞ and A: L_{∞,c}(Ω) → L_{1,loc}(Ω) be a linear operator satisfying

∥ρ_{ε_{0},z}Aρ_{ε_{0},z}^{−1}∥_{p→q} ≤ C (z ∈ Ω′)  (6)

for some C, ε_{0} > 0. Then A extends to consistent bounded operators A_{r} on L_{r}(Ω) (r ∈ [p, q]), and the spectrum σ(A_{r}) is independent of r ∈ [p, q]. In the case q = ∞ the operator A: L_{∞,c}(Ω) → L_{∞}(Ω) is σ(L_{∞,c}, L_{1})-weak^{∗}-continuous; A_{∞} denotes the unique weak^{∗}-continuous extension.

Our second result is restricted to integral operators A acting in all L_{p}-spaces, but we are able to relax the lower volume bound (1) and do not need to consider a superset Ω′ ⊃ Ω. Instead, we require that there exists R > 0 such that

μ(B(x, R)) > 0 for almost all x ∈ Ω.  (7)

This condition is in particular satisfied (even for all R > 0) if (Ω, d) is separable. See also Remark 3(i).

Moreover we are able to adjust our assumptions to the particular volume growth, i.e. to the growth of the function m. This is done by considering a modified metric d_{1} := ψ ∘d where ψ: [0, ∞) → [0, ∞) is an increasing, continuous and subadditive function satisfying ψ(0) = 0. Note that a subadditive function is continuous if and only if it is continuous at 0. We require that

lim inf_{r→∞}(ψ(αr) − ψ(r)) > 0 for some α > 1  (8)

and

n_{1}((x, y) ↦ e^{−ψ(εd(x,y))}) < ∞ (ε > 0).  (9)

The first condition is a lower bound for the growth of ψ at infinity, in particular ψ must grow at least logarithmically. See also Remark 3(ii). The second condition for ε = 1 implies that the volume growth with respect to the metric d_{1} is at most exponential. Observe that, if m(r) is a bound for the volume of the d-balls of radius r in Ω, then m(ψ^{−1}(r)) is a bound for the d_{1}-balls of radius r in Ω. This means that ψ may change the volume growth from subexponential to exponential. See Subsection 2.4 for an example where the volume of balls with respect to d_{1} does grow exponentially.

Under these assumptions we have L_{p}-spectral independence if there is a suitable offdiagonal decay of the kernel of A.

Theorem 2. Assume that (7), (8) and (9) hold. Let A: L_{∞,c}(Ω) → L_{1,loc}(Ω) be a linear operator given by a measurable kernel k: Ω × Ω → ℂ. Let g(x, y) := |k(x, y)| exp(d_{1}(x, y)) and assume that n_{1}(g) < ∞ and n_{p_{0}}(g) < ∞ for some p_{0} > 1. Then, for all p ∈ [1, ∞], A acts as a bounded operator A_{p} on L_{p}(Ω), and A_{∞} is weak^{∗}-continuous. The A_{p} are consistent and the spectrum σ(A_{p}) is independent of p ∈ [1, ∞].

Remark 3. (i) A close inspection of the proof of Theorem 1 below shows that, for (p, q) = (1, ∞), assumption (1) in Theorem 1 may be replaced by (7). But we point out that, for (p, q) = (1, ∞), the weighted norm estimate (6) implies that A has a kernel k satisfying n_{∞}(ke^{ε_{0}d}) < ∞ and n_{1}(ke^{εd}) ≤ n_{1}(e^{−(ε_{0}−ε)d})n_{∞}(ke^{ε_{0}d}) < ∞ for ε < ε_{0}. Therefore the assumptions of Theorem 2 are fulfilled in this case with ψ(r) = r and p_{0} = ∞.

We now show that condition (7) is equivalent to the following condition: There exist R > 0 and a sequence (x_{n}) ⊂ Ω such that

μ(Ω \ ⋃_{n} B(x_{n}, R)) = 0.  (10)

If (10) holds for some R > 0 then obviously (7) holds with 2R in place of R. Assume conversely that (7) holds. Fix x_{0} ∈ Ω. Then, for r > 0, the ball B(x_{0}, r) has finite volume by (viii), and for almost all x ∈ B(x_{0}, r) we have μ(B(x, R)) > 0. An application of Zorn’s Lemma shows that for all r > 0 there exists a sequence (x_{n}(r)) ⊂ Ω such that μ(B(x_{0}, r)\⋃_{n} B(x_{n}(r), 2R)) = 0. It follows that (10) is valid with 2R in place of R since Ω = ⋃_{r∈ℕ} B(x_{0}, r). Notice that condition (10) holds for all R > 0 if (Ω, d) is separable. Further, note that condition (10) cannot be weakened by assuming it for a superset Ω′ ⊃ Ω instead.

(ii) Condition (9) implies that for ε ∈ (0, 1] and almost all x ∈ Ω we have

μ(B(x, r)) ≤ ∫_{Ω} e^{ψ(εr)−ψ(εd(x,y))} dμ(y) ≤ n_{1}(e^{−ψ(εd)})e^{ψ(εr)} (r > 0)  (11)

since ψ is increasing. From this and the subadditivity of ψ it follows that (2) holds for a subexponential function m: [0, ∞) → [0, ∞). If, conversely, (2) holds and for all ε > 0 there exist C > 0, α < 1 such that m(r) ≤ Ce^{αψ(εr)}, then (9) holds. Therefore, in the particular but important case ψ(r) = r condition (9) is equivalent to the validity of (2) and (3).

If we assume ψ to be concave then it is subadditive since ψ(0) = 0. In this case condition (8) is equivalent to the condition ess lim inf_{r→∞} rψ′(r) > 0, see Section 5. This condition clearly holds for the concave functions ψ(r) = r^{α} (with 0 < α ≤ 1) and ψ(r) = α log(1+r) (with α > 0). The latter function is adjusted to the case of polynomial volume growth: If m(r) ≤ c(1 + r)^{N} (r > 0) for some c, N > 0 (e.g. Ω ⊂ ℝ^{N} and μ the Lebesgue measure) and if α > N, then condition (9) is fulfilled for ψ(r) := α log(1 + r). Conversely, if (9) is fulfilled for this function ψ, then (2) holds with m(r) = n_{1}(e^{−d_{1}})(1+r)^{α} by (11). See Subsection 2.4 for an example.

(iii) In Section 5 we show a variant of Theorem 2 which in particular implies the following: Instead of n_{1}(g) < ∞ assume more restrictively that the volume bound (2) holds and that g ≤ h ∘ d for some non-increasing function h: (0, ∞) → [0, ∞] satisfying ∫_{0}^{∞} h(r) dm(r) < ∞ (note that this implies h(0)m(0+) < ∞). Then Theorem 2 is valid without assumption (8).

(iv) The condition n_{1}(g) < ∞ is not needed in the case ψ(r) = r, i.e. d_{1} = d: Let ψ̃(r) := r/2, then (8) and (9) are obviously fulfilled for ψ̃ in place of ψ. Setting ρ(x, y) := exp(−d(x, y)/2) and g̃ := |k| exp(ψ̃ ∘ d) = g · ρ we further have n_{p_{0}}(g̃) ≤ n_{p_{0}}(g) < ∞ and n_{1}(g̃) ≤ n_{p_{0}}(g) · n_{p′_{0}}(ρ) < ∞ by Hölder’s inequality.

(v) Observe that the spaces L_{1,loc}(Ω) and L_{∞,c}(Ω) depend on the choice of the sequence (Ω_{n}). The special choice of (Ω_{n}) may depend on the problem under consideration. For instance, one may want to choose the sets Ω_{n} also bounded with respect to another measurable (semi-)metric d_{0} on Ω′. To simply consider the space of all functions which are integrable over every d-bounded set instead of L_{1,loc}(Ω) and the space of all bounded functions with support essentially contained in a d-bounded set instead of L_{∞,c}(Ω) may not always be the most convenient choice. See Subsection 2.3 for an example.

(vi) By the spectral mapping theorem for the resolvent (see, e.g., [5, Lemma 2.11]) we have independence of the spectrum σ(A_{r}) for r ∈ [p, q] if the A_{r} are unbounded operators in L_{r}(Ω) such that for some λ ∈ ⋂_{r∈[p,q]} ρ(A_{r}) the resolvent operators R := R(λ, A_{r}) are consistent and the assumptions of one of our theorems hold for R: L_{∞,c}(Ω) → L_{1,loc}(Ω).

This is of special interest in the case that the A_{r} are generators of consistent C_{0}- semigroups T_{r}(t) on L_{r}(Ω). Then the consistency of the resolvents R(λ, A_{r}) holds for λ larger than the maximum of the types of T_{p} and T_{q}. See Subsections 2.1 and 2.3 for applications.

(vii) If the measure space (Ω, μ) is finite, then it is known that in (6) no weights are needed (see [1, Prop. 1.1]) for the assertion of Theorem 1. This follows also from our result if we take d = 0 as a semi-metric on Ω′ = Ω. Observe that (1) holds since B(z, R) = Ω for all R > 0.

(viii) The space (Ω, μ) must be assumed to be σ-finite since otherwise condition (2) cannot hold: We show that condition (2) implies an upper bound for the volume of all balls in Ω′. Let x ∈ Ω′. If μ(B(x, r)) > 0 then (2) implies that there exists z ∈ B(x, r) satisfying μ(B(z, 2r)) ≤ m(2r). It follows that μ(B(x, r)) ≤ m(2r) for all x ∈ Ω′. A simple example shows that this bound cannot be improved.

This paper is organized as follows. In Section 2 we discuss the relation of our results to previous ones on the problem of L_{p}-spectral independence. In Section 3 we prove Theorem 1, and in Section 4 we prove Theorem 2. In Section 5 we discuss the assumptions of Theorem 2 and present variations of this result. In Section 6 we present results on pindependence of the spectrum in weighted L_{p}-spaces, making use of several estimates obtained in Section 3 and Section 4.

Acknowledgement. The authors would like to thank Jürgen Voigt for carefully reading an early version of the manuscript and for a number of very valuable remarks.

2 Comments and examples

In this section we relate our results to the existing literature on L_{p}-spectral independence and we present some examples to indicate the improvements.

The case studied most is (Ω, μ) where Ω ⊂ ℝ^{N} is an open subset and μ is the Lebesgue measure. Endowed with the Euclidean distance d conditions (2) and (3) hold. Choosing a suitable superset Ω′ of Ω if necessary, also (1) holds (take, e.g., a δ-neighbourhood of Ω, or simply ℝ^{N}).

In this setting, p-independence of the L_{p}-spectrum has been proved for the generator A of consistent C_{0}-semigroups in L_{p}(Ω, μ) under the assumption that the semigroup operators are given by kernels which satisfy Gaussian bounds. Making use of Remark 3(vi) we will study this case in Subsection 2.1 below in a more general setting. The first proof of L^{p}-spectral independence making use of Gaussian bounds was carried out in [9] for Schrödinger operators with negative part of the potential in the Kato class. In the general case of validity of Gaussian bounds, L^{p}-spectral independence has been proved in [1] under the assumption that ρ(A_{2}) is connected, for selfadjoint A_{2} in [7] and under certain commutator estimates in [10]. In [13] those additional assumptions were removed. It is known that Gaussian bounds hold for large classes of uniformly elliptic operators (see, e.g., [2], [3], [4], [6], [19], only to mention a few).

Weighted norm estimates may be used to prove L_{p}-spectral independence if the semigroup does not act on all L_{p}-spaces, 1 ≤ p < ∞. In [17] this was done for Schrödinger operators with form bounded negative part of the potential. In [8] weighted norm estimates were used to prove L_{p}-spectral independence for higher order elliptic operators. The ideas from [17] were put in a more general context in [16], assuming so-called generalized Gaussian bounds which involve weight functions x ↦ exp(ξ · x), ξ ∈ ℝ^{N}. Those weight functions correspond to the Euclidean metric in ℝ^{N}. In [18] more general weight functions were used in order to study selfadjoint elliptic operators with unbounded coefficients. There the weights are x ↦ exp(ξφ(x)) where φ is a so-called L_{1}-regular function on ℝ^{N}. This corresponds to the semi-metric d(x, y) := |φ(x) − φ(y)| on ℝ^{N}. In [15], L_{p}-spectral independence was proved for closed (not necessarily selfadjoint) operators assuming a weighted norm estimate for a single resolvent. In Subsection 2.2 we discuss properties of L_{1}-regular functions and show that Theorem 1 of the present paper extends the result of [15, Thm. 1].

We want to emphasize that, until now, all proofs of L_{p}-spectral independence assuming weighted norm estimates relied on the “box method” where ℝ^{N} is split up into congruent cubes Q_{j} and one works in spaces l_{r}(L_{p}(Q_{j})). In contrast, our proof of Theorem 1 below does not use the box method but relies on Lemma 12 as a substitute. Indeed, working in a general measure space which carries a semi-metric it is not clear what one should use instead of the partition into cubes of equal size. In Subsection 2.3 below we give an example of a strictly elliptic operator which illustrates the limitations of cube partitions even in Euclidean space.

A modified metric for (Ω, μ) was already used in [14] where L_{p}-spectral independence was proved for integral operators whose kernels satisfy a certain estimate. As in our Theorem 2 the metric had the form d(x, y) = ψ(|x−y|) where ψ:[0, ∞) → [0, ∞) is increasing, subadditive, continuous at 0 and satisfies ψ(0) = 0. A slight variant of Theorem 2 is an extension of the main result in [14] (see Section 5). Moreover, Theorem 2 gives an answer to [14, Problem 4.9(iii)]. In Subsection 2.4 we give classes of integral operators in ℝ where our Theorems 2 or 1 apply but the result of [14] is not applicable.

Finally, in Subsection 2.5 we show how Theorem 1 may be applied to certain nonintegral regularizable operators where the result of [15] is not applicable.

2.1 Gaussian bounds in metric spaces

In this subsection we discuss a result of Davies [7] which stimulated us to study the problem of p-independence in the context of (semi-)metric spaces. The result is as follows:

Let (Ω, d) be a separable metric space, μ a Borel measure on Ω satisfying

μ(B(x, r)) ≤ \{ \begin{matrix}c_{0}r^{N} & if 0 ≤ r ≤ 1 \\ c_{0}r^{M} & if 1 ≤ r < ∞ \end{matrix}  (12)

for all x ∈ Ω and r > 0 for some constants c_{0} > 0 and 0 < N ≤ M < ∞. Let H be a non-negative selfadjoint operator in L_{2}(Ω). Assume that the generated semigroup e^{−tH} on L_{2}(Ω) is given by an integral kernel K(t, x, y) satisfying the Gaussian upper bound

|K(t, x, y)| ≤ c_{1}t^{−N/2} exp(−c_{2} d(x, y)^{2}/t)

for all t > 0 and x, y ∈ Ω for some positive constants c_{1}, c_{2}.

Then the semigroup e^{−tH} on L_{1}(Ω)∩L_{∞}(Ω) extends to consistent C_{0}-semigroups T_{p}(t) on L_{p}(Ω) for all 1 ≤ p < ∞. Let −H_{p} denote the generator of T_{p}(t). Then the spectrum of H_{p} is independent of p ∈ [1, ∞).

Note that in the general context of metric spaces the Gaussian upper bound is much less restrictive than what it seems: Assume that for d(x, y) ≥ 1 we only have

|K(t, x, y)| ≤ c_{1}t^{−N/2} exp(−c_{2} d(x, y)^{γ}/t) (t > 0)

for some 0 < γ < 1. Then just define another metric d_{1} := ψ ∘d on Ω with the subadditive function

ψ(r) := \{ \begin{matrix}r & if r ≤ 1 \\ r^{γ/2} & if r ≥ 1 \end{matrix} .

Replacing d by d_{1}, the Gaussian upper bound is trivially fulfilled, and the volume bound (12) holds with 2M/γ in place of M.

Davies’ result is restricted to the symmetric case. In the following we are going to show that without the assumption of symmetry Theorem 2 can be applied to the resolvent R(λ, H) for λ large enough. This will be a consequence of the following

Lemma 4. Let m_{0}(r) := r^{N}e^{r} (r ≥ 0) for some N > 0 and dm_{0} be the corresponding measure on [0, ∞). Let α > 0. Then

t^{−N/2}∥r ↦ e^{−αr^{2}/t}∥_{L_{p}(dm_{0})} ≤ Mt^{− \frac{N}{2p′}} e^{\frac{t}{pα}} (1 ≤ p ≤ ∞, 0 < t < ∞)

for some M > 0.

Proof. For p = ∞ the estimate obviously holds with M = 1. We are going to prove it for p = 1, then the claim follows by interpolation. We have to show that

I := ∫_{0}^{∞} t^{−N/2}e^{−αr^{2}/t}dm_{0}(r) ≤ Me^{t/α} (0 < t < ∞).

There exists c > 0 with dm_{0}(r) = (1 + r/N)e^{r}dr^{N} ≤ ce^{\sqrt{2}r}dr^{N}. Using the elementary inequality r ≤ \frac{t}{\sqrt{2}α} + \frac{αr^{2}}{2\sqrt{2}t} we obtain I ≤ ∫_{0}^{∞} t^{−N/2}e^{−(1/2)αr^{2}/t}ce^{t/α}dr^{N}. Substituting r = \sqrt{t}s we deduce

I ≤ ∫_{0}^{∞} e^{−(1/2)αs^{2}}ds^{N} · ce^{t/α} ≤ Me^{t/α}

and conclude the proof.

Now assume that (2) holds with m(r) ≤ c_{0}r^{N} for r ≤ 1. Let ψ: [0, ∞) → [0, ∞) be an increasing subadditive function with ψ(r) = r (r ≤ 1) and ψ(r) ≥ log \frac{m(r)}{c_{0}} (r ≥ 1). Then m ∘ ψ^{−1} ≤ c_{0}m_{0}. As in the introduction d_{1} := ψ ∘ d is a metric on Ω and m(ψ^{−1}(r)) is a bound for the d_{1}-balls of radius r in Ω. As in (5) we obtain that, for any non-increasing function φ: [0, ∞) → [0, ∞) and p ∈ [1, ∞], we can estimate

n_{p}(φ ∘ d_{1}) ≤ ∥φ∥_{L_{p}(d(m∘ψ^{−1}))} ≤ c_{0}∥φ∥_{L_{p}(dm_{0})}.

Let T (t) be a C_{0}-semigroup on L_{2}(Ω) given by an integral kernel K(t, x, y) satisfying the Gaussian upper bound

|K(t, x, y)| ≤ c_{1}t^{−N/2} exp(−c_{2} d_{1}(x, y)^{2}/t)  (13)

for all t > 0 and x, y ∈ Ω for some positive constants c_{1}, c_{2}. Then ∥T (t)↾_{L_{1}(Ω)∩L_{∞}(Ω)}∥_{p→p} ≤ c_{1}n_{1}(t^{−N/2} exp(−c_{2} d_{1}^{2}/t)) ≤ Me^{t/c_{2}} by (4) and the above lemma. Hence the semigroup T (t) on L_{1}(Ω)∩L_{∞}(Ω) extends to consistent semigroups T_{p}(t) on L_{p}(Ω) for all 1 ≤ p ≤ ∞. As in [7] one shows that these semigroups are strongly continuous for p < ∞ (for p > 1 this is clear by interpolation). Let −A_{p} denote the generator of T_{p}(t).

By the inequality r ≤ \frac{t}{2α} + \frac{αr^{2}}{2t}, we have |K(t, x, y)|e^{d_{1}(x,y)} ≤ c_{1}t^{−N/2} exp(\frac{t}{2c_{2}} − \frac{c_{2}d_{1}(x,y)^{2}}{2t} ). For λ > λ_{0} := 1/(2c_{2}) define h_{λ}(r) := c_{1} ∫_{0}^{∞} e^{(λ_{0}−λ)t}t^{−N/2} exp(−\frac{c_{2}r^{2}}{2t} ) dt. By Lemma 4 we conclude

∥h_{λ}∥_{L_{p}(dm_{0})} ≤ c_{1} ∫_{0}^{∞} e^{(λ_{0}−λ)t}t^{−N/2}∥r ↦ e^{− \frac{c_{2}r^{2}}{2t}} ∥_{L_{p}(dm_{0})} dt ≤ c_{1} ∫_{0}^{∞} e^{(λ_{0}−λ)t}Mt^{− \frac{N}{2p′}} e^{\frac{2t}{pc_{2}}} dt.

Note that h_{λ} is a non-increasing function. For p′ > N/2 and λ > λ_{p} := λ_{0} + \frac{2}{pc_{2}} it follows that the resolvent R(λ, A_{2}) is given by an integral kernel k_{λ} satisfying g(x, y) := |k_{λ}(x, y)|e^{d_{1}(x,y)} ≤ h_{λ}(d_{1}(x, y)) and hence n_{p}(g) ≤ c_{0}∥h_{λ}∥_{L_{p}(dm_{0})} < ∞. In the same way g ≤ (h_{λ} ∘ ψ) ∘ d and ∫_{0}^{∞} h_{λ} ∘ ψ(r) dm(r) = ∥h_{λ}∥_{L_{1}(d(m∘ψ^{−1}))} < ∞. Now we can apply Theorem 2 in the variant of Remark 3(iii) to obtain

Proposition 5. Let K and ψ be as above, i.e. (13) holds and (2) holds with m ∘ ψ^{−1} ≤ c_{0}m_{0} for some c_{0} > 0. Then for λ ≥ \frac{3}{c_{2}} the resolvent (λ + A_{2})^{−1} is given by an integral kernel k_{λ} satisfying n_{p}(k_{λ}e^{d_{1}}) < ∞ for all p < \frac{N}{N−2} (p < ∞ if N = 2, p ≤ ∞ if N < 2). If ψ is continuous and satisfies condition (9), then the spectrum of A_{p} is independent of p ∈ [1, ∞).

This proposition in particular covers the case of usual Gaussian upper bounds (i.e., the case ψ(r) = r) if the volume grows uniformly subexponentially. With slight modifications the same can be shown if the semigroup kernel satisfies a Gaussian upper bound of order m.

In [12, Sec. 6] it was observed that, for semigroup generators A, p-independence of the spectrum of R(λ, A_{p})^{n} for some n ∈ ℕ and all large λ implies p-independence of σ(A_{p}). If, instead of (13), the kernels K(t, x, y) only satisfy the upper bound

|K(t, x, y)| ≤ c_{1}t^{−M/2} exp(−c_{2}d_{1}(x, y)^{2}/t) (t > 0, x, y ∈ Ω)  (14)

for some c_{1}, c_{2} > 0 and some M > N, then t^{n−1} exp(−εt)K(t, x, y) satisfies (13) if n ≥ 1 + (M − N)/2 and ε > 0. The arguments leading to Proposition 5 show that R(λ, A_{2})^{n} has a kernel satisfying the assumptions of Theorem 2 in the variant of Remark 3(iii). Hence, under the assumptions of Proposition 5, we get p-independence of the spectrum of R(λ, A_{p})^{n} for all large λ. The observation in [12] then shows that σ(A_{p}) does not depend on p.

Another way of proving L_{p}-spectral independence for operators satisfying (14) is to apply Theorem 1 for Ω′ = ℝ^{N} and the Euclidean metric: Let I be the interval of all p ∈ [1, ∞] such that a bound ∥T (t)∥_{p→p} ≤ C_{p}e^{ω_{p}t} holds. From (14) we obtain

∥T (t)∥_{1→∞,ε} ≤ c_{1}t^{−M/2}e^{μ_{1}ε^{2}t} (t > 0, ε ≥ 0)

for some μ_{1} > 0. Stein interpolation between this bound and the L_{r}-bound for the semigroup (T (t)) in L_{r} yields

∥T (t)∥_{p→q,ε} ≤ Ct^{−1/2}e^{με^{2}t+ω_{p,q}t} (t > 0, ε ≥ 0)

for all p, q ∈ I satisfying 0 ≤ p^{−1} − q^{−1} ≤ M^{−1}. Integration of this bound yields the weighted norm estimate (6) for a suitable resolvent. By Theorem 1 and Remark 3(vi) we obtain p-independence of the spectrum of the generator for p ∈ I.

Bound (14) was shown in [4] for semigroups generated by uniformly elliptic operators satisfying Neumann or Robin boundary conditions on certain subsets of ℝ^{N}. In the same way we could treat situations where a bound

|K(t, x, y)| ≤ c_{1}t^{−N/2} exp(−c_{2}d_{1}(x, y)^{γ}t^{1−γ})

with γ ∈ (1, 2) holds. This was done in [12] on open subsets of ℝ^{N}.

In [20], K. T. Sturm showed that the spectrum of uniformly elliptic operators on an N-dimensional complete Riemannian manifold with Ricci curvature bounded below is pindependent if the volume grows uniformly subexponential in the following sense: Instead of (2) the stronger bound

μ(B(x, r)) ≤ m(r)μ(B(x, 1)) (x ∈ M, r > 0),

holds with a function m satisfying (3). This bound is stronger since sup_{x∈M} μ(B(x, 1)) < ∞ by Bishop’s comparison principle, but inf_{x∈M} μ(B(x, 1)) > 0 does not hold in general. As a matter of fact, Proposition 5 cannot be used to reprove that result. One of the difficulties is that one only has a weaker Gaussian upper bound for the corresponding semigroup. This problem can be solved by resorting to weighted L_{p}-spaces as described in Section 6. The volume growth bound (2) still holds for the weighted measure, but not with m(r) ≤ c_{0}r^{N} (r ≤ 1) in general. Nevertheless it is possible to use Theorem 2 directly to prove p-independence of the spectrum of elliptic operators on Riemannian manifolds ( cf. [22]).

2.2 Weighted norm estimates and L^{1}-regular functions

In this subsection we present a result on L_{p}-spectral independence for closed operators acting in L_{p}(Ω) for some open set Ω ⊂ ℝ^{N} which was proved in [15]. Choosing an appropriate semi-metric d on ℝ^{N} we show that this result is also a consequence of Theorem 1 of the present paper. Before stating the result we need to introduce the following notion which is due to Semenov ([18]): We say that a function φ: ℝ^{N} → ℝ^{N} is L^{1}-regular, if it is Lipschitz continuous (i.e. uniformly Lipschitz continuous) and

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

Theorem ([15], Theorem 1). Given 1 ≤ p < q < ∞ let T_{p} and T_{q} be closed operators in L_{p}(Ω) and L_{q}(Ω), respectively. If there exist ε > 0, C < ∞, λ_{0} ∈ ρ(T_{p}) ∩ ρ(T_{q}) and an L^{1}-regular function φ: ℝ^{N} → ℝ^{N} such that

(i) R(λ_{0}, T_{p}), R(λ_{0}, T_{q}) are consistent,

(ii) ∥e^{ξφ}R(λ_{0}, T_{p})e^{−ξφ}↾_{L_{∞,c}(Ω)}∥_{p→q} ≤ C for all ξ ∈ ℝ^{N}, |ξ| ≤ ε,

then σ(T_{p}) = σ(T_{q}), and R(λ, T_{p}), R(λ, T_{q}) are consistent for all λ ∈ ρ(T_{p}) = ρ(T_{q}).

In [15], this theorem was applied to (not necessarily selfadjoint) strictly elliptic operators in divergence form with lower order terms. We now show that, for the semi-metric on Ω′ := ℝ^{N} defined by d(x, y) := |φ(x) − φ(y)|_{∞} and for μ =: | · | the Lebesgue measure, assumptions (1) to (3) and the weighted norm estimate (6) hold. The lower volume estimate (1) is a direct consequence of the Lipschitz continuity of φ: Let L > 0 such that |φ(x) − φ(y)|_{∞} ≤ L|x − y|_{∞} for all x ∈ ℝ^{N}. Then B(x, r) ⊃ x + [−r/L, r/L]^{N} and hence |B(x, r)| ≥ (\frac{2}{L})^{N} r^{N} for all r > 0.

The following lemma in particular shows that (2) and (3) hold.

Lemma 6. Let φ: ℝ^{N} → ℝ^{N} be Lipschitz continuous. Define the semi-metric d on ℝ^{N} as above. Then the following are equivalent:

(i) φ is L_{1}-regular,

(ii) n_{1}(e^{−εd}) < ∞ for all ε > 0,

(iii) there exists C > 0 such that |B(x, r)| ≤ C(1 + r)^{N} for all r > 0.

Proof. (iii)⇒(ii) is shown in the introduction.

(ii)⇒(i) For j ∈ ℤ^{N} let Q_{j} := j + [−\frac{1}{2}, \frac{1}{2}]^{N}. Let L be as above. Then for k ∈ ℤ^{N} we have

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

Since φ is Lipschitz continuous the function x ↦ ∫_{ℝ^{N}} e^{−d(x,y)} dy is continuous, and we conclude

sup_{k∈ℤ^{N}} ∑_{j∈ℤ^{N}} e^{−|φ(k)−φ(j)|} ≤ e^{L/2} sup_{k∈ℝ^{N}} ∫_{ℝ^{N}} e^{−d(k,y)} dy = e^{L/2}n_{1}(e^{−d}) < ∞.

(i)⇒(iii) (cf. [15, Appendix A]) Let r > 0 and let n ∈ ℕ with n − 1 < r ≤ n. Then

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

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

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

|φ^{−1}(Q)| ≤ #{j ∈ ℤ^{N}; d(j, k) ≤ L + 1} ≤ ∑_{j∈ℤ^{N}} e^{L+1−d(j,k)}.

By the L_{1}-regularity of φ this show that |φ^{−1}(Q)| can be estimated from above independently of the cube Q, and the proof is finished.

To conclude, we show that the weighted norm estimate (ii) of the above theorem with the unbounded weights e^{ξφ} implies estimate (6) with the bounded weights ρ_{ε,z} = e^{−εd(·,z)}. To this end let E := {±εe_{j}; j = 1, …, N} where e_{j} are the standard unit vectors of ℝ^{N}. Fix z ∈ ℝ^{N} and let ρ_{ξ} := e^{ξ(φ−φ(z))} for ξ ∈ E. Then

ρ_{ε,z}^{−1} = e^{ε|φ−φ(z)|_{∞}} = max_{ξ∈E} ρ_{ξ}^{−1}.

For f ∈ L_{∞,c}(Ω) and the dual operator A′ of A (cf. Lemma 10 of Section 3) we obtain

∥ρ_{ε,z}^{−1}A′ρ_{ε,z}f∥_{p′} ≤ ∑_{ξ∈E} ∥ρ_{ξ}^{−1}A′ρ_{ξ}ρ_{ξ}^{−1}ρ_{ε,z}f∥_{p′} ≤ ∑_{ξ∈E} ∥ρ_{ξ}^{−1}A′ρ_{ξ}∥_{q′→p′}∥ρ_{ξ}^{−1}ρ_{ε,z}f∥_{q′}.

Noting ρ_{ξ}^{−1}ρ_{ε,z} ≤ 1, we conclude by Lemma 10(ii) and the definition of ρ_{ξ}:

∥ρ_{ε,z}Aρ_{ε,z}^{−1}∥_{p→q} = ∥ρ_{ε,z}^{−1}A′ρ_{ε,z}∥_{q′→p′} ≤ ∑_{ξ∈E} ∥ρ_{ξ}^{−1}A′ρ_{ξ}∥_{q′→p′} ≤ 2N max_{ξ∈E} ∥e^{ξφ}Ae^{−ξφ}∥_{p→q}.

Therefore (6) is fulfilled and we can apply Theorem 1, taking into account Remark 3(vi).

2.3 Elliptic operators in Euclidean space

We now want to discuss in some detail the fact that even in Euclidean space it is very helpful to have the possibility to resort to a totally different semi-metric and also very helpful not to be restricted to “cube partitions”.

Let Ω ⊂ ℝ^{N} be an open set, μ the Lebesgue measure on Ω. Let a_{jk} ∈ L_{1,loc}(Ω) be real-valued such that a_{jk} = a_{kj} for all 1 ≤ j, k ≤ N and

∑_{j,k=1}^{N} a_{jk}ξ_{j}ξ_{k} ≥ α|ξ|^{2} a.e. (ξ ∈ ℝ^{N})

for some α > 0. We define the differential operator −∇·(a∇) with zero Dirichlet boundary conditions by the form method. Let τ be the following sesquilinear form in L^{2}:

τ(u, v) := ∫_{Ω} ∑_{j,k=1}^{N} a_{jk}(x)∂_{k}u(x)∂_{j}v̄(x) dx, D(τ) := C_{c}^{∞}(Ω).

It is well-known that the form τ is closable and that τ̄ is a Dirichlet form. Let H be the selfadjoint operator in L_{2} associated with τ̄. Then e^{−tH}↾_{L_{∞,c}(Ω)} extends to a C_{0}-semigroup e^{−tH_{p}} on L_{p} for all 1 ≤ p < ∞. Here L_{∞,c}(Ω) means the space corresponding to the sequence (Ω_{n}) := ({x ∈ Ω; |x| ≤ n, dist(x, ∂Ω) ≥ 1/n}) where dist denotes the Euclidean distance. This is the appropriate choice for later application of [15, Thm. 8].

If H is uniformly elliptic, i.e. the coefficients a_{jk} are bounded, the semigroup e^{−tH} satisfies an upper Gaussian estimate, and σ(H_{p}) is independent of p ∈ [1, ∞) ([1, Example 5.2]).

In [18], Semenov studied elliptic operators which are not necessarily uniformly elliptic. In this case the semigroup may not satisfy an upper Gaussian estimate (with respect to the Euclidean metric), but under certain conditions p-independence of the spectrum still holds. He showed the same if the operator is perturbed by a real-valued potential with form small negative part. Then the semigroup does not necessarily exist in all L_{p}-spaces.

In [15], the conditions from [18] were slighty relaxed, and an additional first order perturbation was included. This required a new method of proof since the resulting operator need no longer be selfadjoint.

We are now going to present an example in dimension N = 2 which is not covered by the above mentioned results but is by our Theorem 1: Let Ω := Ω′ := {x ∈ ℝ^{2}; x_{1} > 0, 0 < x_{2} < x_{1}^{−2}} ∪ {x ∈ ℝ^{2}; x_{1} ≤ 0, 0 < x_{2} < 2}. Note that Ω has infinite measure, but the unbounded subset Ω_{0} := Ω ∩ {x; x_{1} ≥ 0, x_{2} ≤ 2} has finite measure.

Let a_{jk} be as above and assume that a_{22}(x) ≤ c_{0}x_{2} for x_{2} > 2 and a_{11}(x) ≤ c_{0} for x_{1} < 0. By

d(x, y) := |\sqrt{x_{2} ∨ 2} − \sqrt{y_{2} ∨ 2}| + |x_{1} ∧ 0 − y_{1} ∧ 0| (x, y ∈ Ω)

(where ∨ denote the maximum, ∧ the minimum) we define a semi-metric on Ω. We now consider open balls of radius r with respect to this semi-metric. Obviously such a ball contains Ω_{0} if and only if it contains a point of Ω_{0}. A direct calculation shows that 4r ≤ μ(B(x, r)) ≤ 4r + 2\sqrt{2} for all x ∈ Ω, r > 0. Hence conditions (1) to (3) are fulfilled for the metric d, choosing m(r) := 4r + 2\sqrt{2} (r > 0) and m(0) := 0.

Observe that in this example the space of all essentially bounded functions with support in a d-bounded set is strictly larger than the space L_{∞,c}(Ω) we have chosen since it contains the characteristic function of Ω_{0} (cf. Remark 3(v)). On the other hand, the space of all functions that are integrable on all d-bounded sets is strictly smaller than L_{1,loc}(Ω).

In order to show the weighted estimate (6) for the resolvents of H, we want to apply [15, Thm. 8]. For 0 < ε ≤ 1, z ∈ Ω and ρ := ρ_{ε,z} we obtain

ρ^{−2} ∑_{j,k=1}^{2} a_{jk}∂_{k}ρ∂_{j}ρ = ∑_{j,k=1}^{2} a_{jk}ε^{2}∂_{k}d(·, z)∂_{j}d(·, z)
= ε^{2}(a_{11}(∂_{1}d(·, z))^{2}χ_{{x;x_{1}<0}} + a_{22}(∂_{2}d(·, z))^{2}χ_{{x;x_{2}>2}}) ≤ c_{0}.

Further ρ ∧ n and ρ^{−1} ∧ n are Lipschitz continuous for all n ∈ ℕ, so the conditions of [15, Thm. 8] are fulfilled. We conclude that for 1 < p < q < ∞ there exist λ ∈ ℝ, C > 0 such that

∥ρ_{ε,z}(λ + H)^{−1}ρ_{ε,z}^{−1}↾_{L_{∞,c}(Ω)}∥_{p→q} ≤ C (0 < ε ≤ 1, z ∈ Ω).

By Remark 3(vi), it follows that the spectrum of H_{p} is independent of p ∈ (1, ∞). In fact the same is true for p = 1, but this is not covered by the above mentioned theorem.

Applying [15, Thm. 2] directly to this situation would lead to stronger conditions on the coefficients a_{jk}: On the one hand, the linear growth a_{22} would not be allowed, on the other hand there would be an additional condition on a_{11} on the subset Ω_{0}. This is clear from the discussion on L_{1}-regular functions in the previous subsection.

2.4 Integral operators

The following two propositions present classes of kernel operators which may be treated by the results of this paper but not necessarily by the result in [14] which treated the case φ(y) = y. As in Section 1 let ψ: [0, ∞) → [0, ∞) be increasing and subadditive with ψ(0) = 0. Then d_{1}(x, y) := ψ(|x − y|) (x, y ∈ ℝ) defines a metric on ℝ.

Proposition 7. Let k: ℝ^{2} → ℂ be measurable such that

|k(x, y)| ≤ h(x − φ(y)) (x, y ∈ ℝ),

where h and φ satisfy the following assumptions: h exp(ψ(|·|)) ∈ L_{1} ∩L_{p_{0}} for some p_{0} > 1 and φ: ℝ → ℝ is measurable such that s := sup_{t} |φ(t) − t| < ∞. Let A be the integral operator on L_{∞,c}(ℝ) with kernel k. Then there exists a constant C > 0 such that

∥e^{εd_{1}(·,z)}Ae^{−εd_{1}(·,z)}∥_{1→p_{0}} ≤ C

for all ε ∈ [0, 1] and all z ∈ ℝ.

Proof. For all ε ∈ [0, 1], z, y ∈ ℝ we have

∫ |e^{εd_{1}(x,z)}k(x, y)e^{−εd_{1}(y,z)}|^{p_{0}} dx ≤ ∫ e^{εp_{0}d_{1}(x,y)}h(x − φ(y))^{p_{0}} dx
≤ e^{εp_{0}d_{1}(y,φ(y))} ∫ e^{εp_{0}d_{1}(x,φ(y))}h(x − φ(y))^{p_{0}} dx
≤ e^{p_{0}ψ(s)}∥e^{ψ(|·|)}h∥_{p_{0}}^{p_{0}},

which yields the claim with C := exp(ψ(s))∥ exp(ψ(| · |))h∥_{p_{0}}.

Assume that ψ(r)/ log r → ∞ (r → ∞). Then conditions (1) to (3) hold for the balls with respect to the metric d_{1} and we can apply Theorem 1 to the operator A from Proposition 7. Since k need not decay exponentially as |x − y| → ∞, we may not apply the result of [15], cf. the end of Subsection 2.5.

Proposition 8. Assume in addition to the assumptions of Proposition 7 that φ ∈ C^{1} is strictly increasing and that (φ^{−1})′ is bounded. Then we have n_{p_{0}}(|k|e^{d_{1}}) < ∞.

Proof. We interchange the roles of x and y in the previous proof and use the substitution y = φ^{−1}(η) to obtain

∫ |k(x, y)e^{d_{1}(x,y)}|^{p_{0}} dy ≤ ∫ e^{p_{0}d_{1}(x,y)}h(x − φ(y))^{p_{0}} dy
= ∫ e^{p_{0}d_{1}(x,φ^{−1}(η))}h(x − η)^{p_{0}}(φ^{−1})′(η) dη
≤ e^{p_{0}ψ(s)}∥(φ^{−1})′∥_{∞}∥e^{ψ(|·|)}h∥_{p_{0}}^{p_{0}}

which together with the previous proof implies n_{p_{0}}(|k|e^{d_{1}}) < ∞.

If the conditions of Proposition 8 are fulfilled and ψ is continuous and satisfies conditions (8) and (9), we can apply Theorem 2. Here we in particular have in mind the function ψ(r) := α log(1 + r) for some α > 1. Then (9) is fulfilled since e^{ψ(εr)} = (1 + εr)^{α} and ∫_{0}^{∞}(1 + εr)^{−α}dr < ∞ for all ε > 0. The function g in Theorem 2 is g = |k|(1 + d)^{α} in this case. Further observe that the volume of a d_{1}-ball with radius r is 2ψ^{−1}(r) = 2(e^{r/α} − 1). Hence (ℝ, d_{1}) has exponential volume growth.

We now give an example of a kernel k satisfying the assumptions of Proposition 7 but not of Proposition 8. Let α ∈ (0, 1) and define h by h(x) := |x|^{−α} if |x| ≤ 1 and 0 otherwise. Let β > 1/α and define φ by φ(y) := y if |y| ≥ 1 and φ(y) := |y|^{β} sgn y if |y| < 1. Define the kernel k by k(x, y) := h(x − φ(y)) and denote the corresponding integral operator by A. Let ψ(r) = r. Then A satisfies the assumptions of Proposition 7 for all p_{0} < 1/α and we can apply Theorem 1.

But A does not satisfy the assumptions of Theorem 2 (in particular it does not satisfy the assumptions of the result in [14]) since n_{1}(k) = ∞: We have ∫_{−1}^{1} |y|^{−αβ} dy = ∞ by the assumption β > 1/α. By the monotone convergence theorem we conclude

∥k(x, ·)∥_{1} = ∫_{ℝ} h(x − φ(y)) dy ≥ ∫_{−1/2}^{1/2}|x − |y|^{β} sgn y|^{−α} dy ≥ ∫_{−1/2}^{1/2}(|x| + |y|^{β})^{−α} dy → ∞

as x → 0. This implies n_{1}(k) = ∞.

In the following we show that A is even not L_{r}-bounded for r > \frac{β−1}{αβ−1}. To this aim let 1 − αβ < γ < 1 and define f: ℝ → ℝ by f(y) := |y|^{−γ} if |y| ≤ 1 and 0 otherwise. Then (γ −1)/β > −α and hence δ := α+(γ −1)/β > 0. Since α, γ < 1 we get δ −α+1 ∈ (0, 1). Using this and the substitution z = |y|^{β} sgn y it is elementary to show that Af(x) ∼ |x|^{−δ} as x → 0. Hence Af ∉ L_{r}(ℝ) for r^{−1} ≤ δ, but f ∈ L_{r}(ℝ) for γ < r^{−1}. It remains to show that for r > \frac{β−1}{αβ−1} there exists 1 − αβ < γ < 1 such that γ < r^{−1} ≤ α + (γ − 1)/β. A direct calculation yields that γ := r^{−1}β + 1 − αβ fulfils these inequalities. If, e.g., α = 3/4 and β = 3, then taking γ = 1/4 shows that A is not L_{2}-bounded.

2.5 Non-integral regularizing operators

According to [21], for all 1 < p < q < ∞ with p^{−1} − q^{−1} < 1/2, there exists a linear nonintegral operator A: L_{p}(ℝ) → L_{q}(ℝ), given by convolution with a certain finite positive Borel measure ν with supp ν ⊂ [0, 1]. Let ψ be a subadditive increasing function satisfying ψ(0) = 0 and ψ(r)/ log r → ∞ as r → ∞. By d(x, y) := ψ(|x − y|) we define a metric on ℝ. Then conditions (1) to (3) hold for the balls with respect to the metric d if we endow ℝ with the Lebesgue measure.

For ε_{0} = 1 and C := e^{ψ(1)}∥A∥_{p→q}, estimate (6) holds, since

|e^{−d(x,z)}(A e^{d(·,z)}f)(x)| ≤ e^{−d(x,z)} ∫_{0}^{1} e^{d(z,x−y)}|f(x − y)| dν(y)
≤ ∫_{0}^{1} e^{ψ(|y|)}|f(x − y)| dν(y) ≤ e^{ψ(1)}(A|f|)(x) (x, z ∈ ℝ)

for all f ∈ L_{∞,c}(ℝ). Now let k be a measurable function on ℝ^{2} satisfying |k(x, y)| ≤ exp(−2ψ(|x − y|)) (x, y ∈ ℝ) and define S to be the integral operator with kernel k.

Then we can apply Theorem 1 to A + S but none of the theorems requiring a pointwise defined kernel; in particular, we cannot apply the result of [14].

Now assume that exp(−cψ(|x|) − cψ(|y|)) ≤ |k(x, y)| ≤ exp(−2ψ(|x − y|)) for some c ≥ 2. Using Lemma 6 from Subsection 2.2, we are going to show that we cannot apply the result of [15] to A + S if ψ(r)/r → 0 as r → ∞, e.g. ψ(r) = \sqrt{r}. To this end suppose there exist ε > 0 and an L_{1}-regular function φ: ℝ → ℝ satisfying

∥e^{ξφ}(A + S)e^{−ξφ}∥_{p→q} < ∞ for all ξ ∈ [−ε, ε].

Without loss of generality assume that φ(0) = 0. Let 0 ≤ f ∈ L_{∞,c}(ℝ), f ≠ 0. Then

e^{ξφ}(A + S)e^{−ξφ}f(x) ≥ e^{ξφ}Ae^{−ξφ}f(x) + e^{ξφ(x)−cψ(|x|)} ∫_{ℝ} e^{−cψ(|y|)−ξφ(y)}f(y) dy.

for all ξ ∈ [−ε, ε]. The first summand on the right hand side is in L_{∞,c}(ℝ). Suppose the second is in L_{q}(ℝ) for some q ∈ [1, ∞] and ξ = ±ε. Then in particular sup_{x∈ℝ} (ε|φ(x)| − cψ(|x|)) < ∞, i.e., |φ(x)| ≤ cψ(|x|)/ε + k for some k > 0. Since φ(0) = 0, Lemma 6 implies that

|\{y ∈ ℝ; ψ(|y|) ≤ \frac{ε}{c}(r − k)\}| ≤ |\{y ∈ ℝ; |φ(0) − φ(y)| ≤ r\}| ≤ C(1 + r) (r > k).

This contradicts the assumption ψ(r)/r → 0 as r → ∞.

3 Proof of Theorem 1

We start with a series of preparatory results. For this whole section we fix 1 ≤ p ≤ q ≤ ∞.

In order to prove the inclusion ρ(A_{r}) ⊂ ρ(A_{s}) for r, s ∈ [p, q] one has to show that for λ ∈ ρ(A_{r}) the operator R(λ, A_{r})↾_{L_{∞,c}(Ω)} extends to a bounded operator on L_{s}(Ω). This is expressed in the following elementary lemma which is stated in the general context of topological spaces (cf. [15, Prop. 4] or [1, Prop. 2.3]).

Let E, F, G be Hausdorff spaces with E, F ↪ G such that E ∩ F is dense in both E and F. Let D ⊂ E ∩ F be a subset which is dense with respect to the initial topology coming from the embeddings E ∩ F ↪ E and E ∩ F ↪ F.

Lemma 9. Let A_{E}: E → E and A_{F}: F → F be continuous mappings satisfying A_{E}↾_{D} = A_{F} ↾_{D}. Assume that A_{E} is continuously invertible and A_{E}^{−1}↾_{D} extends to a continuous mapping R: F → F. Then A_{F} is continuously invertible, and A_{F}^{−1} = R.

Proof. Since D is dense in E ∩ F and E, F ↪ G, we have A_{E}↾_{E∩F} = A_{F} ↾_{E∩F} and A_{E}^{−1}↾_{E∩F} = R↾_{E∩F}. Hence RA_{F} = A_{F} R = I on E ∩ F. The density of E ∩ F in F yields the claim.

To apply this lemma we further need the following. Consider the dual system ⟨L_{∞,c}(Ω), L_{1,loc}(Ω)⟩ and endow L_{∞,c}(Ω) and L_{1,loc}(Ω) with the corresponding weak topologies. If A:L_{∞,c}(Ω) → L_{1,loc}(Ω) is a weakly continuous operator, then A has a dual operator A′: L_{∞,c}(Ω) → L_{1,loc}(Ω), i.e. ∫ Af · g = ∫ f · A′g for all f, g ∈ L_{∞,c}(Ω).

Lemma 10. Let A: L_{∞,c}(Ω) → L_{1,loc}(Ω) be a linear operator, ∥A∥_{p→q} < ∞.

(i) If p < ∞, then A is weakly continuous.

(ii) If A is weakly continuous, then for the dual operator A′ one has ∥A′∥_{q′→p′} = ∥A∥_{p→q}.

(iii) If A is weakly continuous and p = q = ∞, then A has a unique weak^{∗}-continuous extension A_{∞}: L_{∞}(Ω) → L_{∞}(Ω).

Proof. (i) Let A_{p}: L_{p}(Ω) → L_{q}(Ω) be the continuous extension of A. Then A_{p} is weakly continuous. Since L_{p}(Ω) is reflexive, A_{p} is weak^{∗}-continuous. This implies (i).

(ii) follows from ∥A∥_{p→q} = sup{| ∫ Af · g|; f, g ∈ L_{∞,c}(Ω), ∥f∥_{p} = ∥g∥_{q′} = 1}.

(iii) By (ii) we have ∥A′∥_{1→1} < ∞. Let Ã: L_{1}(Ω) → L_{1}(Ω) denote the continuous extension of A′. Then A_{∞} := Ã′ is the unique weak^{∗}-continuous extension of A.

Corollary 11. Let A_{r}: L_{r}(Ω) → L_{r}(Ω) be consistent bounded operators (p ≤ r ≤ q), A_{∞} weak^{∗}-continuous if q = ∞. If λ ∈ ρ(A_{r}) and ∥R(λ, A_{r})↾_{L_{∞,c}(Ω)}∥_{s→s} < ∞ for some r, s ∈ [p, q], then λ ∈ ρ(A_{s}).

Proof. For r, s < ∞ the result directly follows from Lemma 9 with G = L_{1,loc}(Ω) and D = L_{∞,c}(Ω).

We endow L_{∞}(Ω) with the weak^{∗}-topology. Then, for t < ∞, L_{∞,c}(Ω) is dense in L_{t}(Ω) ∩ L_{∞}(Ω), and L_{t}(Ω) ∩ L_{∞}(Ω) is dense in L_{t}(Ω) and L_{∞}(Ω).

For the case r = ∞ it remains to note that λ − A_{∞} has a bounded inverse if and only if it has a weak^{∗}-continuous inverse. In the case r < s = ∞ the assumptions imply that R(λ, A_{r})↾_{L_{∞,c}(Ω)} is L_{r}- and L_{∞}-bounded. Hence, by Lemma 10(i) and (iii), R(λ, A_{r})↾_{L_{∞,c}(Ω)} has a weak^{∗}-continuous extension R: L_{∞}(Ω) → L_{∞}(Ω), and again we are done by Lemma 9.

Now we construct a norm equivalent to the L_{q}-norm using the weights ρ_{ε,z}. Here is the only point where assumption (1) is needed.

Lemma 12. For all f ∈ L_{q}(Ω) and ε > 0 we have

∥f∥_{q,ε} := ∥z ↦ ∥ρ_{ε,z}f∥_{L_{q}(Ω)}∥_{L_{q}(Ω′)} ≥ c_{q,ε}^{−1}∥f∥_{q},

where c_{q,ε} := γ^{−1/q}e^{εR}.

Proof. By Fubini’s theorem we obtain ∥f∥_{q,ε} = ∥x ↦ ∥ρ_{ε,x}∥_{L_{q}(Ω′)}f(x)∥_{L_{q}(Ω)}. Assumption (1) implies ∥ρ_{ε,x}∥_{L_{q}(Ω′)} ≥ ∥e^{−εR}χ_{B(x,R)}∥_{L_{q}(Ω′)} ≥ e^{−εR}γ^{1/q} for almost all x ∈ Ω. This gives the desired conclusion.

The following consequence of (4) will be used throughout:

∥Ω′ ∋ z ↦ ∥g(z, ·)f∥_{p}∥_{q} ≤ n_{t}(g)∥f∥_{s} (p ≤ s, t ≤ q, p^{−1} + q^{−1} = s^{−1} + t^{−1}), (15)

where g is a measurable function on Ω′ × Ω′, f a measurable function on Ω. In particular, for p = q, g = ρ_{ε} we have ∥f∥_{q,ε} ≤ n_{q}(ρ_{ε})∥f∥_{q}, so ∥ · ∥_{q,ε} is indeed equivalent to ∥ · ∥_{q}.

For a linear operator A: L_{∞,c}(Ω) → L_{1,loc}(Ω) and ε ∈ ℝ we will use the following notation:

∥A∥_{p→q,ε} := sup_{z∈Ω′} ∥ρ_{ε,z}Aρ_{ε,z}^{−1}∥_{p→q}
= inf{c > 0; ∀f ∈ L_{∞,c}(Ω), z ∈ Ω′ : ∥ρ_{ε,z}Af∥_{q} ≤ c∥ρ_{ε,z}f∥_{p}} ∈ [0, ∞].

With this notation assumption (6) reads ∥A∥_{p→q,ε_{0}} ≤ C.

Using Lemma 12 and the estimate (15) only, we see that assumption (6) implies that the operator A is L_{q}-bounded: Let f ∈ L_{∞,c}(Ω), then

∥Af∥_{q} ≤ c_{q,ε_{0}}∥Af∥_{q,ε_{0}} ≤ c_{q,ε_{0}} ∥z ↦ ∥A∥_{p→q,ε_{0}}∥ρ_{ε_{0},z}f∥_{p}∥_{q} ≤ c_{q,ε_{0}}n_{p}(ρ_{ε_{0}})∥A∥_{p→q,ε_{0}}∥f∥_{q}.

But we are going to establish much more general estimates. For ε > 0, M ≥ 1 we define a class of weight functions

P (ε, M) := {ρ: Ω′ → (0, ∞) measurable; ∀u, x ∈ Ω′ : ρ(x)/ρ(u) ≤ Mρ_{−ε,u}(x)}.

For ρ ∈ P (ε, M), ε′ ∈ ℝ and u ∈ Ω′ we have

ρ_{ε′,u}ρρ(u)^{−1} ≤ Mρ_{ε′−ε,u} and ρ_{ε′,u}ρ(u)ρ^{−1} ≤ Mρ_{ε′−ε,u}.  (16)

Note that due to the triangle inequality ρ_{ε,z} ∈ P (|ε|, 1) for ε ∈ ℝ and z ∈ Ω′. We now estimate several operator norms in terms of ∥A∥_{p→q,ε_{0}} (cf. [17, Prop. 3.2]).

Proposition 13. (i) Let p ≤ s ≤ q, t^{−1} := p^{−1} + q^{−1} − s^{−1}. For 0 ≤ ε < ε_{0}, M ≥ 1 and ρ ∈ P (ε, M) we have

∥ρAρ^{−1}∥_{s→q} ≤ M^{2}c_{q,ε_{0}+ε}n_{t}(ρ_{ε_{0}−ε})∥A∥_{p→q,ε_{0}}.

(ii) There exist constants C_{ε_{0},ε} < ∞, bounded as ε → 0, such that

∥A∥_{s→t,ε} ≤ C_{ε_{0},ε}∥A∥_{p→q,ε_{0}}

for all p ≤ s ≤ t ≤ q and |ε| < ε_{0}.

Proof. We begin with (i). Let ε′ := ε_{0} + ε and f ∈ L_{∞,c}(Ω). By Lemma 12 we have

∥ρAρ^{−1}f∥_{q} ≤ c_{q,ε′} ∥u ↦ ∥ρ_{ε′,u}ρρ(u)^{−1}Aρ(u)ρ^{−1}f∥_{q}∥_{q}.

Using (16) and the definition of ∥A∥_{p→q,ε_{0}} we get

∥ρ_{ε′,u}ρρ(u)^{−1}Aρ(u)ρ^{−1}f∥_{q} ≤ M∥ρ_{ε_{0},u}Aρ(u)ρ^{−1}f∥_{q} ≤ M∥A∥_{p→q,ε_{0}}∥ρ_{ε_{0},u}ρ(u)ρ^{−1}f∥_{p}
≤ M^{2}∥A∥_{p→q,ε_{0}}∥ρ_{ε_{0}−ε,u}f∥_{p}.

By (15) we conclude

∥ρAρ^{−1}f∥_{q} ≤ c_{q,ε′}M^{2}∥A∥_{p→q,ε_{0}} ∥u ↦ ∥ρ_{ε_{0}−ε,u}f∥_{p}∥_{q} ≤ M^{2}c_{q,ε′}∥A∥_{p→q,ε_{0}}n_{t}(ρ_{ε_{0}−ε})∥f∥_{s}.

This proves (i).

For t = q, part (ii) is just a special case of (i) since ρ_{ε,z} ∈ P (|ε|, 1) for z ∈ Ω′. Note that the constant C_{ε_{0},ε} does not depend on p, q, s, t since c_{q,ε′} ≤ c_{1,ε′} ∨ c_{∞,ε′} and n_{q}(ρ_{ε_{0}−ε}) ≤ n_{1}(ρ_{ε_{0}−ε}) ∨ n_{∞}(ρ_{ε_{0}−ε}). Further note that the constants stay bounded as ε → 0.

We now prove (ii) for s = t = p. Note that the case p = ∞ is treated above, so we assume p < ∞. By Lemma 10(i) we know that ρ_{ε_{0},z}Aρ_{ε_{0},z}^{−1}: L_{∞,c}(Ω) → L_{1,loc}(Ω) is weakly continuous for all z ∈ Ω′. It easily follows that A is weakly continuous and that (ρ_{ε,z}Aρ_{ε,z}^{−1})′ = ρ_{ε,z}^{−1}A′ρ_{ε,z} for all ε ∈ ℝ, z ∈ Ω′. Hence by Lemma 10(ii) and by (i) we obtain

∥A′∥_{q′→p′,ε_{1}} = ∥A∥_{p→q,−ε_{1}} ≤ c_{q,ε_{0}+ε_{1}}n_{q}(ρ_{ε_{0}−ε_{1}})∥A∥_{p→q,ε_{0}}

for ε_{0} > ε_{1} > 0. For |ε| < ε_{0}, choosing |ε| < ε_{1} < ε_{0} and using (i) again for the dual situation we conclude

∥A∥_{p→p,ε} = ∥A′∥_{p′→p′,−ε}
≤ c_{p′,ε_{1}+|ε|}n_{q′}(ρ_{ε_{1}−|ε|})∥A′∥_{q′→p′,ε_{1}}
≤ c_{p′,ε_{1}+|ε|}n_{q′}(ρ_{ε_{1}−|ε|})c_{q,ε_{0}+ε_{1}}n_{q}(ρ_{ε_{0}−ε_{1}})∥A∥_{p→q,ε_{0}}.

Now Riesz-Thorin interpolation between the cases treated above proves the remaining cases.

Remark 14. Let A: L_{∞,c}(Ω) → L_{1,loc}(Ω) be weakly continuous with ∥A∥_{p→q,ε_{0}} < ∞ for some ε_{0} > 0. By Lemma 10(i) the latter implies weak continuity if — as in the assumptions of Theorem 1 — p < ∞.

(i) Proposition 13(ii) yields that A extends to a bounded operator A_{r} on L_{r}(Ω) for r ∈ [p, q], r < ∞. In the case q = ∞ Lemma 10(iii) implies that A has a unique weak^{∗}-continuous extension A_{∞}: L_{∞}(Ω) → L_{∞}(Ω). The operators A_{r} are consistent since L_{∞,c}(Ω) is dense in L_{r}(Ω) ∩ L_{s}(Ω) if L_{∞}(Ω) is endowed with the weak^{∗}-topology.

(ii) If ρ ∈ P (ε, M) for some ε < ε_{0}, M ≥ 1, then ρAρ^{−1} extends to a bounded operator A_{ρ}: L_{q}(Ω) → L_{q}(Ω) by Proposition 13(i). In the case q = ∞, again A_{ρ}: L_{∞}(Ω) → L_{∞}(Ω) is the unique weak^{∗}-continuous extension. In case ρ = ρ_{ε,z} for some |ε| < ε_{0}, z ∈ Ω′ we write A_{ε,z} for A_{ρ_{ε,z}}.

(iii) If an operator A is defined on D(A) ⊃ L_{∞,c}(Ω), we also write ∥A∥_{p→q,ε} instead of ∥A↾_{L_{∞,c}(Ω)}∥_{p→q,ε}. If A: L_{p}(Ω) → L_{q}(Ω) is bounded (weak^{∗}-continuous in case p = ∞), then ∥ρ_{ε,z}Af∥_{q} ≤ ∥A∥_{p→q,ε}∥ρ_{ε,z}f∥_{p} for all f ∈ L_{p}(Ω). Therefore, if B: L_{q}(Ω) → L_{r}(Ω) is a bounded operator (weak^{∗}-continuous in case q = ∞), then ∥BA∥_{p→r,ε} ≤ ∥A∥_{p→q,ε}∥B∥_{q→r,ε}.

The crucial part in the proof of Theorem 1 is the following estimate which implies convergence of weighted operators (cf. [16, Lemma 3.2.3] or [15, Prop. 5(iii)]).

Proposition 15. There exist δ_{ε_{0},ε} > 0 with δ_{ε_{0},ε} → 0 (ε → 0) such that

∥ρAρ^{−1} − A∥_{s→t} ≤ δ_{ε_{0},ε}∥A∥_{p→q,ε_{0}}

for all p ≤ s ≤ t ≤ q, 0 ≤ ε < ε_{0} and ρ ∈ P (ε, 1).

Proof. First note that ∥A∥_{s→t,ε_{1}} ≤ C_{ε_{0},ε_{1}}∥A∥_{p→q,ε_{0}} for all 0 < ε_{1} < ε_{0} by Proposition 13(ii). Hence it suffices to treat the case s = p, t = q. Using Lemma 12, for f ∈ L_{∞,c}(Ω) we have

∥(ρAρ^{−1} − A)f∥_{q} ≤ c_{q,ε_{0}+ε}∥u ↦ ∥ρ_{ε_{0}+ε,u}(ρAρ^{−1} − A)f∥_{q}∥_{q}.  (17)

We now write

ρAρ^{−1} − A = ρρ(u)^{−1}A(ρ(u)ρ^{−1} − 1) + (ρρ(u)^{−1} − 1)A,

insert this into (17), use the triangle inequality and estimate the two resulting terms separately. For the second we have, using (16) and (15),

∥u ↦ ∥ρ_{ε_{0}+ε,u}(ρρ(u)^{−1} − 1)Af∥_{q}∥_{q} ≤ n_{q}(ρ_{ε_{0}+ε}(ρ_{−ε} − 1))∥A∥_{p→q}∥f∥_{p}.

Using (16) and (15) again, the first term can be estimated by

∥u ↦ ∥ρ_{ε_{0}+ε,u}ρρ(u)^{−1}A(ρ(u)ρ^{−1} − 1)f∥_{q}∥_{q} ≤ ∥u ↦ ∥ρ_{ε_{0},u}A(ρ(u)ρ^{−1} − 1)f∥_{q}∥_{q}

≤ ∥A∥_{p→q,ε_{0}} ∥u ↦ ∥ρ_{ε_{0},u}(ρ_{−ε,u} − 1)f∥_{p}∥_{q}

≤ ∥A∥_{p→q,ε_{0}}n_{q}(ρ_{ε_{0}}(ρ_{−ε} − 1))∥f∥_{p}.

The claim follows after noting that ∥A∥_{p→q} ≤ C_{ε_{0},0}∥A∥_{p→q,ε_{0}} and

n_{q}(ρ_{ε_{0}}(ρ_{−ε} − 1)) ≤ n_{q}(ρ_{ε_{0}/2}) sup_{r≥0} e^{−ε_{0}r/2}(e^{εr} − 1) → 0 (ε → 0),

where we used ρ_{ε_{0}} = ρ_{ε_{0}/2}^{2}.

The following consequence of Proposition 15 will be used in the proof of Theorem 1.

Corollary 16. Let A be an operator on L_{q}(Ω) satisfying ∥A∥_{q→q,ε_{0}} < ∞ for some ε_{0} > 0. In case q = ∞ assume that A is weak^{∗}-continuous. Then for all λ ∈ ρ(A) there exists ε > 0 such that λ ∈ ρ(A_{ε,z}) for all z ∈ Ω′, and ∥R(λ, A)∥_{q→q,ε} < ∞.

Proof. Let λ ∈ ρ(A). By assumption we have ∥λ − A∥_{q→q,ε_{0}} < ∞. Applying Proposition 15 and recalling that inversion is continuous in the open set of invertible elements in L(L_{q}(Ω)), we have for some ε > 0 that λ − A_{ε,z} is invertible for all z ∈ Ω′ (that is, the first claim), and that

sup_{z∈Ω′} ∥R(λ, A_{ε,z})∥_{q→q} < ∞.  (18)

To prove ∥R(λ, A)∥_{q→q,ε} < ∞ observe that for all z ∈ Ω′ the function ρ_{ε,z}^{−1}:D(ρ_{ε,z}^{−1}, L_{q}) → L_{q} is surjective since ρ_{ε,z} is bounded. Hence R(λ, A_{ε,z})↾_{D(ρ_{ε,z}^{−1},L_{q})} = ρ_{ε,z}R(λ, A)ρ_{ε,z}^{−1} and we are done by (18).

Proof of Theorem 1. In Remark 14(i) it is shown that A extends to consistent bounded operators A_{r} on L_{r}(Ω) (p ≤ r ≤ q, A_{∞} weak^{∗}-continuous if q = ∞). So we need to prove the inclusion ρ(A_{r}) ⊂ ρ(A_{s}) for all r, s ∈ [p, q]. Let λ ∈ ρ(A_{r}).

First we treat the case λ ≠ 0. Then we can rewrite the resolvent R(λ) of A_{r} as follows

R(λ) = λ^{−1}I + λ^{−2}A_{r} + λ^{−2}A_{r}R(λ)A_{r}.

We have to show that λ ∈ ρ(A_{s}) which by Corollary 11 amounts to showing L_{s}- boundedness of R(λ)↾_{L_{∞,c}(Ω)}.

It is clear that λ^{−1}I +λ^{−2}A is L_{s}-bounded; we will show that A_{r}R(λ)A is L_{s}-bounded. According to Corollary 16 we have ∥R(λ)∥_{r→r,ε} < ∞ for some 0 < ε < ε_{0}. By Proposition 13(ii) ∥A∥_{p→r,ε} + ∥A∥_{r→q,ε} < ∞. Remark 14(iii) implies ∥A_{r}R(λ)A∥_{p→q,ε} < ∞. Another application of Proposition 13(ii) gives L_{s}-boundedness of A_{r}R(λ)A.

In the case λ = 0 we simply write R(λ) = A_{r}R(λ)^{3}A_{r}, hence by the above we have ∥R(λ)∥_{p→q,ε} < ∞. Again ∥R(λ)∥_{s→s} < ∞ by Proposition 13(ii).

Remark 17. Let A_{p} be a bounded operator on L_{p}(Ω) with 0 ∈ ρ(A_{p}) and ∥A_{p}∥_{p→q} < ∞. Then ∥I: L_{p}(Ω) → L_{q}(Ω)∥ ≤ ∥A_{p}^{−1}∥_{p→p}∥A_{p}∥_{p→q} < ∞. Therefore (Ω, μ) cannot contain a sequence (M_{n}) of subsets satisfying M_{n} ⊃ M_{n+1} (n ∈ ℕ) and 0 < μ(M_{n}) → 0 (n → ∞).

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

4 Proof of Theorem 2

For the proof of Theorem 2 we use a different type of weight functions ρ_{ε,z} which we define for ε ∈ [0, 1] and z ∈ Ω by ρ_{ε,z}(x) := exp(−d_{ε}(x, z)) where d_{ε}(x, z) := ψ(εd(x, z)). Note that this is consistent with the definition of d_{1} in the introduction and that each d_{ε} is a semi-metric (and d_{0} ≡ 0), since ψ is increasing and subadditive, ψ ≥ 0 and ψ(0) = 0.

Let A:L_{∞,c}(Ω) → L_{1,loc}(Ω) be a linear operator given by a measurable kernel k:Ω×Ω → ℂ. Let g(x, y) := |k(x, y)| exp(d_{1}(x, y)) and assume that n_{1}(g) < ∞ and n_{p_{0}}(g) < ∞ for some p_{0} > 1. We begin with some preparatory results.

Proposition 18. If 1 ≤ p ≤ q ≤ ∞ and 1 + q^{−1} − p^{−1} ≥ p_{0}^{−1}, we have for all ε ∈ [0, 1]

∥A∥_{p→q,ε} := sup_{z∈Ω} ∥ρ_{ε,z}Aρ_{ε,z}^{−1}∥_{p→q} < ∞.

Proof. The operator ρ_{ε,z}Aρ_{ε,z}^{−1} is given by the kernel

k_{ε,z}(x, y) := exp(−d_{ε}(x, z))k(x, y) exp(d_{ε}(y, z)),  (19)

so by (4) we only have to show n_{r}(k_{ε,z}) < ∞ for r^{−1} := 1 + q^{−1} − p^{−1}. For ε ∈ [0, 1] we have the estimate

|k_{ε,z}(x, y)| ≤ exp(d_{ε}(x, y))|k(x, y)| ≤ g(x, y)

since ψ is increasing. Taking into account that 1 ≤ r ≤ p_{0}, the assumption n_{1}(g)+n_{p_{0}}(g) < ∞ implies the assertion.

Proposition 19. For all p ∈ [1, ∞] we have

sup_{z∈Ω} ∥ρ_{ε,z}Aρ_{ε,z}^{−1} − A∥_{p→p} → 0 (ε → 0).

Proof. The operators ρ_{ε,z}Aρ_{ε,z}^{−1} − A are given by kernels m_{ε,z} = k_{ε,z} − k with k_{ε,z} from (19). For x, y, z ∈ Ω and ε ∈ (0, 1) we have

|m_{ε,z}(x, y)| ≤ (e^{−d_{ε}(x,z)+d_{ε}(y,z)} − 1)|k(x, y)| ≤ e^{−d_{1}(x,y)}(e^{d_{ε}(x,y)} − 1)g(x, y). (20)

Hence |m_{ε,z}| ≤ sup_{r} e^{−ψ(r)}(e^{ψ(εr)} − 1) · g which implies n_{1}(m_{ε,z}) ≤ sup_{r} e^{−ψ(r)}(e^{ψ(εr)} − 1) · n_{1}(g). The proof is complete after we prove the following lemma.

Lemma 20. Let ψ: [0, ∞) → [0, ∞) be increasing with ψ(r) → 0 as r → 0. Then (8) is equivalent to

sup_{r≥0} e^{−ψ(r)}(e^{ψ(εr)} − 1) → 0 (ε → 0).

Proof. We have sup_{r≥0} e^{−ψ(r)}(e^{ψ(εr)}−1) = sup_{r≥0} e^{−ψ(r/ε)}(e^{ψ(r)}−1). First we show that (8) implies sup_{0≤r≤r_{0}} e^{−ψ(r/ε)}(e^{ψ(r)} − 1) → 0 as ε → 0 for all r_{0} > 0. This follows from the estimate

e^{−ψ(r/ε)}(e^{ψ(r)} − 1) ≤ \{ \begin{matrix}1 · (e^{ψ(\sqrt{ε})} − 1) , & 0 ≤ r ≤ \sqrt{ε} \\ e^{−ψ(1/\sqrt{ε})}(e^{ψ(r_{0})} − 1) , & \sqrt{ε} ≤ r ≤ r_{0} \end{matrix}

since ψ(r) → 0 as r → 0 and ψ(r) → ∞ as r → ∞. Now e^{ψ(r)} ≥ e^{ψ(r_{0})} > 1 for all r ≥ r_{0} > 0, so there is a constant c > 0 such that ce^{ψ(r)} ≤ e^{ψ(r)} − 1 ≤ e^{ψ(r)} for all r ≥ r_{0}. Hence sup_{r≥r_{0}} e^{−ψ(r/ε)}(e^{ψ(r)} − 1) → 0 as ε → 0 is equivalent to

c(ε, r_{0}) := inf_{r≥r_{0}}(ψ(r/ε) − ψ(r)) → ∞ (ε → 0).

This in turn holds for some r_{0} > 0 if and only if (8) holds.

Proof of Theorem 2. The proof differs only slightly from the proof of Theorem 1. By Proposition 18 (with ε = 0) A acts as a bounded operator A_{p} on L_{p}(Ω) (1 ≤ p ≤ ∞). The operators A_{p} are consistent since they are all defined by the same integral kernel. Let r ∈ [1, ∞] and λ ∈ ρ(A_{r}). First we assume that λ ≠ 0.

Fix n ∈ ℕ such that 1−p_{0}^{−1} ≥ n^{−1}. Proposition 18 then gives ∥A_{r}^{n}∥_{1→r,ε}+∥A_{r}^{n}∥_{r→∞,ε} < ∞. For the resolvent R(λ) of A_{r} we have

R(λ) = ∑_{j=1}^{2n} λ^{−j}A_{r}^{j−1} + λ^{−2n}A_{r}^{n}R(λ)A_{r}^{n}.  (21)

As in Section 3 we have to show L_{s}-boundedness of R(λ)↾_{L_{∞,c}(Ω)} for all s ∈ [1, ∞].

This is clear for ∑_{j=1}^{2n} in (21); we will show it for A_{r}^{n}R(λ)A_{r}^{n}. As in the proof of Theorem 1 let A_{ε,z} denote the (weak^{∗}-)continuous extension of ρ_{ε,z}Aρ_{ε,z}^{−1} to L_{r}(Ω). Applying Proposition 19 and recalling that inversion is continuous in the open set of invertible elements in L(L_{r}(Ω)) we have for some ε > 0 that λ ∈ ρ(A_{ε,z}, L_{r}) for all z ∈ Ω, and that

sup_{z∈Ω′} ∥R(λ, A_{ε,z})∥_{r→r} < ∞.

Since again R(λ, A_{ε,z}) = ρ_{ε,z}R(λ)ρ_{ε,z}^{−1} this implies c := ∥A_{r}^{n}R(λ)A_{r}^{n}∥_{1→∞,ε} < ∞. By the Dunford-Pettis theorem it follows that A_{r}^{n}R(λ)A_{r}^{n} is given by a kernel h and that, for all z ∈ Ω,

|h(x, y)| ≤ c exp(d_{ε}(x, z) − d_{ε}(y, z)) (almost all x, y ∈ Ω).

Choose R > 0 and a sequence (x_{n}) according to (10). Then, for almost all (x, y) ∈ B(x_{n}, R) × Ω, we have

|h(x, y)| ≤ c exp(d_{ε}(x, x_{n}) − d_{ε}(y, x_{n})) ≤ c exp(2ψ(εR) − d_{ε}(x, y)).  (22)

By (9) we get n_{1}(h) ≤ c exp(2ψ(εR))n_{1}(e^{−d_{ε}}) < ∞. Hence A_{r}^{n}R(λ)A_{r}^{n}↾_{L_{∞,c}(Ω)} is bounded on L_{s}(Ω) for all s ∈ [1, ∞].

In the case λ = 0 we write R(0) = A_{r}^{n}R(0)^{2n+1}A_{r}^{n} and obtain as above that R(λ)↾_{L_{∞,c}(Ω)} is bounded on L_{s}(Ω) for all s ∈ [1, ∞].

5 Variants of Theorem 2

In this section we discuss variants of the assumptions in Theorem 2.

Remark 21. Let ψ: [0, ∞) → [0, ∞) be increasing and continuous with ψ(0) = 0. If ψ is subadditive and a.e. differentiable with ess lim inf_{r→∞} rψ′(r) > 0, then condition (8) holds. If ψ is concave, then the converse is also true.

For the proof of the first claim take γ, r_{0} > 0 such that ψ′(r) ≥ γr^{−1} for almost all r ≥ r_{0}. Then (8) holds for all α > 1 by the estimate

ψ(αr) − ψ(r) ≥ ∫_{r}^{αr} ψ′(η) dη ≥ γ ∫_{r}^{αr} η^{−1} dη = γ(log(αr) − log r) = γ log α (r ≥ r_{0}).

If ψ is concave then ψ′(r) ≥ \frac{ψ(αr)−ψ(r)}{αr−r} for all α > 1 and almost all r > 0. If (8) holds we obtain that ess lim inf_{r→∞} rψ′(r) ≥ \frac{1}{α−1} lim inf_{r→∞}(ψ(αr) − ψ(r)) > 0. Finally notice that if ψ: [0, ∞) → [0, ∞) with ψ(0) = 0 is concave and continuous then it is subadditive and absolutely continuous (but the converse is not true).

Remark 22. If ψ satisfies all the assumptions in Theorem 2 except (8), we still obtain p-independence of the spectrum if the function g = e^{d_{1}}|k| in Theorem 2 satisfies as before n_{1}(g) + n_{p_{0}}(g) < ∞ for some p_{0} > 1 and the additional assumption

lim_{n→∞} n_{1}(g_{n}) = 0 where g_{n}(x, y) := 1_{{d(x,y)≥n}}g(x, y) for n ∈ ℕ.  (23)

The only change in the proof is that in the proof of Proposition 19 the last expression in (20) should now be estimated as follows: Letting h_{ε}(r) := exp(−ψ(r))(exp(ψ(εr)) − 1) we have

|m_{ε,z}(x, y)| ≤ h_{ε}(d(x, y))g(x, y) (x, y ∈ Ω),

and h_{ε}(r) ≤ 1 gives the estimate

n_{1}(m_{ε,z}) ≤ n_{1}(g_{n}) + sup_{r∈[0,n]} h_{ε}(r) · n_{1}(g)

for all n ∈ ℕ. We have sup_{r∈[0,n]} h_{ε}(r) → 0 as ε → 0 which implies lim sup_{ε→0} n_{1}(m_{ε,z}) ≤ n_{1}(g_{n}) for all n ∈ ℕ. The assumption n_{1}(g_{n}) → 0 now gives the assertion of Proposition 19.

Assumption (23) is in particular fulfilled if g(x, y) ≤ h(d(x, y)) for some non-increasing function h: (0, ∞) → [0, ∞) satisfying ∫_{0}^{∞} h(r) dm(r) < ∞, with m from (2). To show this observe that h(r) → 0 as r → ∞ since m(r) → ∞ as r → ∞. As in (5) we obtain

n_{1}(g_{n}) ≤ ess sup_{x∈Ω} ∫_{Ω} h(d(x, y)) ∧ h(n) dμ(y) ≤ ∫_{0}^{∞} h(r) ∧ h(n) dm(r) → 0 (n → ∞)

by dominated convergence. This proves Remark 3(iii).

Now we can reprove the main result of [14]: Let Ω ⊂ ℝ^{N} be open and let Ω̃ := Ω − Ω. Let ψ: [0, ∞) → [0, ∞) be increasing, continuous and subadditive such that e^{−ψ(ε|·|)} ∈ L_{1}(Ω̃) for all ε ∈ (0, 1]. Let k: Ω × Ω → ℂ be a measurable kernel satisfying

|k(x, y)| ≤ h(x − y) (x, y ∈ Ω)  (24)

for a function h that satisfies he^{ψ(|·|)} ∈ L_{1}(Ω̃) ∩ L_{p_{0}}(Ω̃). Then the assertion of Theorem 2 holds.

Defining g and g_{n} as above we have for n ∈ ℕ_{0} (note g = g_{0}!)

g_{n}(x, y) ≤ 1_{{|x−y|≥n}}h(x − y)e^{ψ(|x−y|)} (x, y ∈ Ω).

For n = 0, this implies n_{1}(g) + n_{p_{0}}(g) < ∞ by the assumption on h. Moreover, for n ∈ ℕ,

n_{1}(g_{n}) ≤ ∥1_{{|·|≥n}}he^{ψ(|·|)}∥_{L_{1}(Ω̃)}

and the right hand side clearly tends to 0 as n → ∞ by dominated convergence. Hence the assumptions of the variant of Theorem 2 as stated in Remark 22 hold, and the proof is complete.

6 Weighted L_{p}-spaces

In this section we give sufficient conditions for operators A to have the same spectrum in different weighted L_{p}-spaces. For previous results in this direction see [7] and [13].

A weight is a measurable function φ:Ω → (0, ∞) such that φ, φ^{−1} ∈ L_{∞,loc}(Ω), and we define L_{p}(φ) := {f; φf ∈ L_{p}(Ω)} for all 1 ≤ p ≤ ∞. If p < ∞ then L_{p}(φ) = L_{p}(Ω, φ^{p}μ). In the sequel we shall denote L_{p}(Ω) simply by L_{p}. The multiplication operator φ:L_{p}(φ) → L_{p} is an isometry.

Our first result uses the weighted norm estimate (6) for p = q. Recall from Section 3 the definition of the classes P (ε, M), which contain weight functions φ defined on Ω′. We call a weight function φ: Ω′ → (0, ∞) subexponential if, for all ε > 0, φ ∈ P (ε, M_{ε}) for some M_{ε} > 0.

Theorem 23. Assume that (1), (2) and (3) hold. Let 1 ≤ p < ∞ and A be a bounded operator on L_{p}(Ω) satisfying

∥A∥_{p→p,ε_{0}} < ∞  (25)

for some ε_{0} > 0. Let φ be a subexponential weight function. Then A↾_{L_{p}∩L_{p}(φ)} extends to a bounded operator A_{φ} on L_{p}(φ) and σ(A_{φ}) = σ(A).

Here ∥ · ∥_{p→p,ε} is defined as in Section 3.

Remark 24. (i) The above theorem still holds if p = ∞ and A is weak^{∗}-continuous. For the proof just consider the dual situation in L_{1}.

(ii) We point out that in the above theorem only the weighted L_{p}-estimate (25) but no p − q-smoothing is responsible for the equality of the spectra.

Proof of Theorem 23. By Proposition 13(i), φAφ^{−1}↾_{L_{∞,c}} is L_{p}-bounded, hence A↾_{L_{∞,c}} is L_{p}(φ)-bounded. The first claim follows from the density of L_{∞,c} in L_{p} ∩ L_{p}(φ).

We now show the inclusion ρ(A, L_{p}) ⊂ ρ(A_{φ}, L_{p}(φ)). Let λ ∈ ρ(A, L_{p}). By Corollary 16 we have ∥R(λ)∥_{p→p,ε} < ∞ for some 0 < ε < ε_{0}. Proposition 13(i) yields

∥R(λ, A)↾_{L_{∞,c}(Ω)}∥_{L_{p}(φ)→L_{p}(φ)} = ∥φR(λ, A)φ^{−1}↾_{L_{∞,c}(Ω)}∥_{L_{p}→L_{p}} < ∞.

Lemma 9 implies λ ∈ ρ(A_{φ}, L_{p}(φ)).

For the other inclusion we observe that the operator φA_{φ}φ^{−1} on L_{p} satisfies (25) for ε_{1} := ε_{0}/3 in place of ε_{0}: Since φ is subexponential, we have ρ := ρ_{ε_{1},z}φ ∈ P (ε_{0}/2, M) for all z ∈ Ω′. Applying Proposition 13(i) we get ∥φAφ^{−1}∥_{p→p,ε_{1}} < ∞. Since φ^{−1} is subexponential and φA_{φ}φ^{−1} on L_{p} ∩ L_{p}(φ^{−1}) extends to φAφ^{−1} on L_{p}(φ^{−1}), the first part of the proof shows

ρ(A_{φ}, L_{p}(φ)) = ρ(φA_{φ}φ^{−1}, L_{p}) ⊂ ρ(φAφ^{−1}, L_{p}(φ^{−1})) = ρ(A, L_{p}).

This completes the proof of the theorem.

Our second result deals with the situation of Theorem 2. Since we now use the semimetrics d_{ε} = ψ ∘ (εd) from Section 4, we introduce modified classes of weight functions. Namely, for ε ∈ (0, 1) and M ≥ 1 let

P_{ψ}(ε, M) := {φ: Ω → (0, ∞) measurable; ∀x, y ∈ Ω : φ(x)/φ(y) ≤ Me^{d_{1}(x,y)−d_{1−ε}(x,y)}}.

Remark 25. (i) In case ψ(r) = r this definition coincides with the previous one, i.e. P_{ψ}(ε, M) = P (ε, M).

(ii) In case ψ(r) = α log(1 + r) for some α > 0 it is easy to see that the class P_{ψ}(ε, M) only contains bounded weight functions.

(iii) There exist examples of (Ω, μ), d and functions ψ satisfying (8) and (9) such that ψ(r)/ log r → ∞ (r → ∞), but the volume growth of d_{1}-balls is exponential (Ω = ℝ, μ the Lebesgue measure, ψ(r) := log^{2}(1 + r) and d(x, y) := m^{−1}(2|x − y|) where m(r) := e^{ψ(r)/2} − 1). This means that it is not always possible to reduce the situation to the case ψ(r) = r.

We say that φ is ψ-subordinated if, for all ε > 0, there exists M_{ε} > 0 such that φ ∈ P_{ψ}(ε, M). In case ψ(r) = r the two notions ψ-subordinated and subexponential coincide by Remark 25(i). In the following we shall assume that ψ is concave, not only subadditive. Then for all 0 < ε′ < ε < 1 we have

d_{1} − d_{ε} ≤ d_{1−ε′} − d_{ε−ε′}.  (26)

This follows from

ψ(r) − ψ(εr) = ∫_{εr}^{r} ψ′(s) ds ≤ ∫_{(ε−ε′)r}^{(1−ε′)r} ψ′(s) ds = ψ((1 − ε′)r) − ψ((ε − ε′)r) (r > 0),

where we have used that ψ′ is non-increasing. Conversely, ψ(r) − ψ(εr) ≤ ψ((1 − ε′)r) − ψ((ε − ε′)r) for all r > 0, 0 < ε′ < ε < 1 implies that ψ is concave.

Theorem 26. Assume that ψ is concave and that (7), (8) and (9) hold. Let A be a bounded operator on L_{p} given by a measurable kernel k: Ω × Ω → ℂ. Let g := |k| exp(d_{1}) and assume that n_{1}(g) < ∞ and n_{p_{0}}(g) < ∞ for some p_{0} > 1. Let φ be a ψ-subordinated weight function. Then A acts as a bounded operator A_{φ} on L_{p}(φ) and σ(A_{φ}) = σ(A).

Proof. The assumptions on ψ and on the kernel k of A imply that φAφ^{−1} is L_{p}-bounded; cf. Proposition 18. Hence A↾_{L_{∞,c}(Ω)} is L_{p}(φ)-bounded and the first claim is proved.

We now show the inclusion ρ(A, L_{p}) ⊂ ρ(A_{φ}, L_{p}(φ)). Let λ ∈ ρ(A, L_{p}). We use (21) for the resolvent R(λ) and — as before — we show L_{p}(φ)-boundedness of R(λ)↾_{L_{∞,c}(Ω)}.

It is clear that ∑_{j=1}^{2n} in (21) is L_{p}(φ)-bounded. The operator φA^{n}R(λ)A^{n}φ^{−1} has the kernel h_{1}(x, y) := φ(x)h(x, y)φ(y)^{−1} with h as in the proof of Theorem 2. By (22), the assumption on φ and (26) we have the estimate

|h_{1}| ≤ C_{ε/2}e^{d_{1}−d_{1−ε/2}}c_{1}e^{−d_{ε}} ≤ c_{1}C_{ε/2}e^{−d_{ε/2}}.

By assumption (9) this implies L_{p}-boundedness of φA^{n}R(λ)A^{n}φ^{−1}. Hence A^{n}R(λ)A^{n} is L_{p}(φ)-bounded, and we have shown L_{p}(φ)-boundedness of R(λ). By Lemma 9 we obtain λ ∈ ρ(A_{φ}, L_{p}(φ)).

For the proof of the other inclusion we first show that φA_{φ}φ^{−1} satisfies the assumptions of Theorem 26 for ψ̃(r) := ψ(r/2) in place of ψ. Let d̃_{ε} := ψ̃∘ (εd). The kernel of φA_{φ}φ^{−1} is k̃(x, y) := φ(x)k(x, y)φ^{−1}(y), and we have the estimate

|k̃|ed̃_{1} ≤ Ce^{d_{1}−d_{1/2}}|k|e^{d_{1/2}} = C|k|e^{d_{1}}.

The weight function φ^{−1} is ψ̃-subordinated since, for all ε ∈ (0, 1), we can estimate by (26):

φ^{−1}(x)/φ^{−1}(y) ≤ M_{ε/2}e^{d_{1}(x,y)−d_{1−ε/2}(x,y)} ≤ M_{ε/2}e^{d_{1/2}(x,y)−d_{1/2−ε/2}(x,y)} = M_{ε/2}e^{d̃_{1}(x,y)−d̃_{1−ε}(x,y)}.

Since ψ̃ is concave and satisfies (8) and (9), the first part of the proof shows

ρ(A_{φ}, L_{p}(φ)) = ρ(φA_{φ}φ^{−1}, L_{p}) ⊂ ρ(φAφ^{−1}, L_{p}(φ^{−1})) = ρ(A, L_{p}),

as in the proof of Theorem 23.

References

[1] W. Arendt, Gaussian estimates and interpolation of the spectrum in L^{p}, Differential and Integral Equations 7, 1153–1168 (1994).

[2] P. Auscher, Regularity theorems and heat kernels for elliptic operators, J. London Math. Soc. 54, 284–296 (1996).

[3] P. Auscher, A. McIntosh, and Ph. Tchamitchian, Noyau de la chaleur d’opérateurs elliptiques complexes, Math. Research Letters 1, 35–44 (1994).

[4] D. Daners, Heat kernel estimates for for operators with boundary conditions, to appear in Math. Nachr.

28 Peer C. Kunstmann & H. Vogt

[5] E. B. Davies, One-parameter semigroups, Academic Press, 1980.

[6] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge 1989.

[7] E. B. Davies, L^{p} spectral independence and L^{1} analyticity, J. London Math. Soc. 52, 177–184 (1995).

[8] E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132, 141–169 (1995).

[9] R. Hempel and J. Voigt, The spectrum of a Schrödinger operator on L_{p}(ℝ^{ν}) is p-independent, Comm. Math. Phys. 104, 243–250 (1986).

[10] M. Hieber and E. Schrohe, L^{p} spectral independence of elliptic operators via commutator estimates, Positivity 3, No.3, 259–272 (1999).

[11] K. Jörgens, Lineare Integraloperatoren, Teubner, Stuttgart 1970.

[12] S. Karrmann, Gaussian estimates for second order operators with unbounded coefficients, preprint 2000.

[13] P. C. Kunstmann, Heat kernel estimates and L^{p}-spectral independence of elliptic operators, Bull. London Math. Soc. 31, 345–353 (1999).

[14] P. C. Kunstmann, Kernel estimates and L^{p}-spectral independence of differential and integral operators, to appear in Proceedings OT17 Conference.

[15] V. Liskevich and H. Vogt, On L^{p}-spectra and essential spectra of second order elliptic operators, to appear in J. London Math. Soc.

[16] G. Schreieck, L_{p}-Eigenschaften der Wärmeleitungshalbgruppe mit singulärer Absorption, doctoral dissertation, Shaker Verlag, Aachen, 1996.

[17] G. Schreieck and J. Voigt, Stability of the L_{p}-spectrum of generalized Schrödinger operators with form small negative part of the potential, in ”Functional Analysis” Proc. Essen 1991, Bierstedt, Pietsch, Ruess, Vogt eds., Marcel-Dekker, New York, 1994.

[18] Yu. A. Semenov, Stability of L^{p}-spectrum of generalized Schrödinger operators and equivalence of Green’s functions, Internat. Math. Res. Notices 12, 574–593 (1997).

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

[20] K.-Th. Sturm, On the L^{p} spectrum of uniformly elliptic operators on Riemannian manifolds, J. Funct. Anal. 118, 442–453 (1993).

[21] J. Voigt, Non-integral regularizing operators on L_{p}-spaces, Math. Nachr. 202, 173– 176 (1999).

[22] H. Vogt, Analyticity of semigroups on spaces of exponentially bounded volume, manuscript.