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

Peer C. Kunstmann Mathematisches Institut I Universität Karlsruhe Englerstraße 2 D - 76128 Karlsruhe, Germany peer.kunstmann@math.uni-karlsruhe.de

Hendrik Vogt^{∗} Institut für Analysis Fachrichtung Mathematik Technische Universität Dresden D - 01062 Dresden, Germany hendrhugo@egon ik.vogt@tu-darmstadt resden.de

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. The assumptions are formulated in terms of a measurable semi-metric d on (Ω, μ); the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition.

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 and generators of semigroups that satisfy (generalized) Gaussian bounds — to indicate improvements.

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

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

∗supported by Deutsche Forschungsgemeinschaft

1 Introduction and Main Result

Let (Ω, μ) be a σ-finite measure space and 1 ≤ p < q ≤ ∞. Suppose that A is a linear operator that 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 Corollary 2 and Remark 3(iii) 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 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, namely 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 the latter kind of assumptions, i.e., weighted norm estimates, 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 Lebesgue measure. We shall deal here with an arbitrary σ-finite measure space (Ω, μ). The weighted norm estimates for the operator A 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 our result, 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 that is rather different from the arguments used previously in this context.

In order to make the assumptions in our main result 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 different definition in the context of Riemannian manifolds since in general condition (1) does not hold in that context; see the end of Subsection 2.1.

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 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(ii) for these spaces).

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).  (4)

Note that the ρ_{ε,z} are multiplicators on L_{1,loc}(Ω) and on 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 is our main result; the proof is given in Section 3.

1 Theorem. 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′)  (5)

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.

In [12, Sec. 6] it was observed that, in the case of semigroup generators, 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}). Following the lines of [12, proof of Thm. 1.7] we thus obtain the following application of Theorem 1.

2 Corollary. Assume that (1), (2) and (3) hold. Let 1 ≤ p < q ≤ ∞. Let T_{r}(t) be consistent C_{0}-semigroups on L_{r}(Ω), with generators A_{r} (p ≤ r ≤ q), A_{∞} the weak^{∗} generator of the (weak^{∗}-continuous) semigroup T_{∞}(t) if q = ∞. Suppose that

∥ρ_{ε_{0},z}T_{p}(t)ρ_{ε_{0},z}^{−1}∥_{p→q} ≤ ct^{−α}e^{ωt} (z ∈ Ω′, t > 0)  (6)

for some c, α > 0, ω ∈ ℝ. Then the spectrum σ(A_{r}) is independent of r ∈ [p, q].

3 Remark. (i) In the case p = 1, q = ∞, Theorem 1 and Corollary 2 also hold if condition (1) is replaced by the weaker assumption

there exists R > 0 such that μ(B(z, R)) > 0 for a.e. z ∈ Ω′;  (1’)

see the paragraphs introducing Lemma 9 and Proposition 10.

(ii) Observe that the spaces L_{1,loc}(Ω) and L_{∞,c}(Ω) depend on the choice of the sequence (Ω_{n}). The particular 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.

(iii) 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 (5) holds for A = R↾_{L_{∞,c}(Ω)}. See Subsections 2.2 and 2.3 for applications.

(iv) If the measure space (Ω, μ) is finite, then it is known that in (5) no weights are needed for the assertion of Theorem 1 (see [1, Prop. 1.1]). 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.

(v) 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 the latter bound cannot be improved.)

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 comment on the existing literature on L_{p}-spectral independence and 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 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 Corollary 2, 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 [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 9 as a substitute. Indeed, working in a general measure space that 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.

Before proceeding with the examples, we introduce some notation. Let (Ω′, μ) and d be as in the introduction and assume that (1) to (3) hold. 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 then 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),  (7)

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 ≤ ∞).

For ρ_{ε} as in (4) we have n_{∞}(ρ_{ε}) ≤ 1, and by assumption (3) we obtain

n_{p}(ρ_{ε}) ≤ (∫_{0}^{∞} e^{−εpr} dm(r))^{1/p} < ∞ (1 ≤ p < ∞).  (8)

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].

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{cases}c_{0}r^{N} & if 0 ≤ r ≤ 1, \\ c_{0}r^{M} & if 1 ≤ r < ∞ \end{cases}

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 satisfying the Gaussian upper bound

|K(t, x, y)| ≤ c_{1}t^{−N/2}e^{−c_{2} d(x,y)^{2}/t} (t > 0, x, y ∈ Ω),  (9)

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, ∞).

We now show that this result of Davies can be deduced from Corollary 2, thus removing the assumption that the semigroup is symmetric.

4 Proposition. Assume that (1’), (2) and (3) hold. Let T (t) be a C_{0}-semigroup on L_{2}(Ω) given by an integral kernel K satisfying the Gaussian upper bound (9) for some positive constants c_{1}, c_{2}. Assume that T (t) extends to consistent semigroups T_{p}(t) on L_{p}(Ω) for 1 ≤ p ≤ ∞ (strongly continuous for 1 ≤ p < ∞ and weak^{∗}-continuous for p = ∞). Let A_{p} denote the generator of T_{p}(t) for 1 ≤ p < ∞, and let A_{∞} denote the weak^{∗} generator of T_{∞}(t). Then the spectrum of A_{p} is independent of p ∈ [1, ∞].

If m(r) ≤ c_{0}r^{N} (0 < r ≤ 1) for some c_{0} > 0 then a straightforward computation shows that n_{1}(t^{−N/2} exp(−c_{2} d^{2}/t)) ≤ Me^{t/c_{2}} for all t > 0, for some M > 0. Thus, by (7), T (t) extends to a family of consistent semigroups T_{p}(t) on L_{p}(Ω). As in [7] one shows that these semigroups are strongly continuous for p < ∞ (for p > 1 this is clear by interpolation).

Proof of Proposition 4. By the inequality r ≤ \frac{t}{4c_{2}} + \frac{c_{2}r^{2}}{t}, the kernel k_{z,t} of ρ_{1,z}T (t)ρ_{1,z}^{−1} satisfies

k_{z,t}(x, y) = e^{d(z,x)−d(z,y)}K(t, x, y) ≤ e^{d(x,y)}c_{1}t^{−N/2}e^{−c_{2} d(x,y)^{2}/t} ≤ c_{1}t^{−N/2}e^{\frac{t}{4c_{2}}}

for all x, y ∈ Ω. We thus obtain that (6) is satisfied for Ω′ = Ω, ε_{0} = 1, p = 1 and q = ∞, so the assertion follows from Corollary 2 (see also Remark 3(i)).

With slight modifications, the same can be shown if the semigroup kernel satisfies a Gaussian upper bound of order m or, more generally, an upper bound like

|K(t, x, y)| ≤ c_{1}t^{−M/2} exp(−c_{2}d(x, y)^{γ}/t^{γ−1}) (t > 0, x, y ∈ Ω)  (10)

for some c_{1}, c_{2} > 0, M > 0 and γ > 1. Bounds of this type have been proved in [4], [12] for semigroups generated by certain elliptic operators on open subsets Ω ⊂ ℝ^{N}. In [4], the operator was assumed to be uniformly elliptic, satisfying Neumann or Robin boundary conditions, and (10) was shown with γ = 2 and M ≥ N depending on some weak regularity assumption on Ω. In [12], the bound (10) was shown with M = N and some γ ∈ (1, 2) for elliptic operators with unbounded coefficients in the principle part, satisfying Dirichlet boundary conditions.

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, i.e., our condition (1) is not satisfied.

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} that was proved in [15]. Choosing an appropriate semi-metric d on ℝ^{N} we show that this result is also a consequence of our main theorem. We need to introduce the following notion due to Semenov ([18]): A function φ: ℝ^{N} → ℝ^{N} is called 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)|} < ∞.

We use Theorem 1 to reprove 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 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 (5) 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.

5 Lemma. 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 (8).

(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 that

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 shows 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 (5) 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 7 of Section 3 below) 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 7(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 (5) is fulfilled and we can apply Theorem 1, taking into account Remark 3(iii).

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, 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 that 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 that 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, if we choose 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(ii)). 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 (5) 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).

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

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.

3 Proof of the main result

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 that is dense with respect to the initial topology coming from the embeddings E ∩ F ↪ E and E ∩ F ↪ F.

6 Lemma. 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 that 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).

7 Lemma. 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.

8 Corollary. 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 6 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 7(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 6.

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. Observe that in the case q = ∞ it suffices to assume (1’).

9 Lemma. 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 (7) 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}),  (11)

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 (5) reads ∥A∥_{p→q,ε_{0}} ≤ C.

Using Lemma 9 and the estimate (11) only, we see that assumption (5) implies that the operator A is L_{q}-bounded: For f ∈ L_{∞,c}(Ω),

∥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}.  (12)

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]). Again observe that in the case p = 1, q = ∞ it suffices to assume (1’).

10 Proposition. (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 9 we have

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

Using (12) 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 (11) we conclude that

∥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 7(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 7(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 that

∥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.

11 Remark. 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 7(i) the latter implies weak continuity if — as in the assumptions of Theorem 1 — p < ∞.

(i) Proposition 10(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 7(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 10(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)]).

12 Proposition. 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 10(ii). Hence it suffices to treat the case s = p, t = q. Using Lemma 9, 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}.  (13)

We now write

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

insert this into (13), use the triangle inequality and estimate the two resulting terms separately. For the second we have, using (12) and (11),

∥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 (12) and (11) 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 12 will be used in the proof of Theorem 1.

13 Corollary. Let A be an operator on L_{q}(Ω) satisfying ∥A∥_{q→q,ε_{0}} < ∞ for some ε_{0} > 0. In the 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 12 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} < ∞.  (14)

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 (14).

Proof of Theorem 1. In Remark 11(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 8 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 13 we have ∥R(λ)∥_{r→r,ε} < ∞ for some 0 < ε < ε_{0}. Moreover, ∥A∥_{p→r,ε} + ∥A∥_{r→q,ε} < ∞ by Proposition 10(ii). Remark 11(iii) implies ∥A_{r}R(λ)A∥_{p→q,ε} < ∞. Another application of Proposition 10(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} < ∞, and again ∥R(λ)∥_{s→s} < ∞ by Proposition 10(ii).

14 Remark. (i) 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→∞} < ∞.

(ii) Let φ:Ω′ → (0, ∞) be a subexponential weight function, i.e., for all ε > 0 there exists M > 0 such that \frac{ρ(x)}{ρ(y)} ≤ M exp(εd(x, y)) for all x, y ∈ Ω′. Using Lemma 6, Proposition 10(i) and Corollary 13, one easily proves the following version of Theorem 1 for the weighted space L_{p}(φ) := {f; φf ∈ L_{p}(Ω)}, where 1 ≤ p < ∞.

Assume that (1), (2) and (3) hold. Let A be a bounded operator on L_{p}(Ω) satisfying ∥A∥_{p→p,ε_{0}} < ∞ for some ε_{0} > 0. Then A↾_{L_{p}∩L_{p}(φ)} extends to a bounded operator A_{φ} on L_{p}(φ), and σ(A_{φ}) = σ(A).

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