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Abstract

Let be the class of bounded away from one and infinity functions such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space . We show that if belongs to the Hörmander class with , , then the pseudodifferential operator is bounded on the variable Lebesgue space provided that . Let be the class of variable exponents represented as where , , and . We prove that if slowly oscillates at infinity in the first variable, then the condition

is sufficient for the Fredholmness of on whenever . Both theorems generalize pioneering results by Rabinovich and Samko [23] obtained for globally log-Hölder continuous exponents , constituting a proper subset of .

PDO on Variable Lebesgue Spaces] Pseudodifferential Operators
on Variable Lebesgue Spaces A. Yu. Karlovich]Alexei Yu. Karlovich

I. M. Spitkovsky]Ilya M. Spitkovsky

\@mkboth\shortauthors\shorttitle

[

To Professor Vladimir Rabinovich on the occasion of his 70th birthday

Mathematics Subject Classification (2000). Primary 47G30; Secondary 42B25, 46E30.

Keywords. Pseudodifferential operator, Hörmander symbol, slowly oscillating symbol, variable Lebesgue space, Hardy-Littlewood maximal operator, Fefferman-Stein sharp maximal operator, Fredholmness..

\@xsect

We denote the usual operators of first order partial differentiation on by . For every multi-index with non-negative integers , we write . Further, , and for each vector , define and where stands for the Euclidean norm of .

Let denote the set of all infinitely differentiable functions with compact support. Recall that, given , a pseudodifferential operator is formally defined by the formula

where the symbol is assumed to be smooth in both the spatial variable and the frequency variable , and satisfies certain growth conditions (see e.g. [25, Chap. VI]). An example of symbols one might consider is the class , introduced by Hörmander [12], consisting of with

where and and the positive constants depend only on and .

The study of pseudodifferential operators with symbols in on so-called variable Lebesgue spaces was started by Rabinovich and Samko [23, 24].

Let be a measurable a.e. finite function. By we denote the set of all complex-valued functions on such that

for some . This set becomes a Banach space when equipped with the norm

It is easy to see that if is constant, then is nothing but the standard Lebesgue space . The space is referred to as a variable Lebesgue space.

Lemma 1.1.

(see e.g. [14, Theorem 2.11] or [9, Theorem 3.4.12]) If is an essentially bounded measurable function, then is dense in .

We will always suppose that

(1.1)

Under these conditions, the space is separable and reflexive, and its dual space is isomorphic to , where

(see e.g. [14] or [9, Chap. 3]).

Given , the Hardy-Littlewood maximal operator is defined by

where the supremum is taken over all cubes containing (here, and throughout, cubes will be assumed to have their sides parallel to the coordinate axes). By denote the set of all measurable functions such that (1.1) holds and the Hardy-Littlewood maximal operator is bounded on .

Assume that (1.1) is fulfilled. Diening [7] proved that if satisfies

(1.2)

and is constant outside some ball, then . Further, the behavior of at infinity was relaxed by Cruz-Uribe, Fiorenza, and Neugebauer [5, 6], where it was shown that if satisfies (1.2) and there exists a such that

(1.3)

then . Following [9, Section 4.1], we will say that if conditions (1.2)–(1.3) are fulfilled, then is globally log-Hölder continuous.

Conditions (1.2) and (1.3) are optimal for the boundedness of in the pointwise sense; the corresponding examples are contained in [20] and [5]. However, neither (1.2) nor (1.3) is necessary for . Nekvinda [18] proved that if satisfies (1.1)–(1.2) and

(1.4)

for some and , then . One can show that (1.3) implies (1.4), but the converse, in general, is not true. The corresponding example is constructed in [3]. Nekvinda further relaxed condition (1.4) in [19]. Lerner [15] (see also [9, Example 5.1.8]) showed that there exist discontinuous at zero or/and at infinity exponents, which nevertheless belong to . We refer to the recent monograph [9] for further discussions concerning the class .

Our first main result is the following theorem on the boundedness of pseudodifferential operators on variable Lebesgue spaces.

Theorem 1.2.

Let , , and . If , then extends to a bounded operator on the variable Lebesgue space .

The respective result for and satisfying (1.1)–(1.3) was proved by Rabinovich and Samko [23, Theorem 5.1].

Following [23, Definition 4.5], a symbol is said to be slowly oscillating at infinity in the first variable if

where

(1.5)

for all multi-indices and . We denote by the class of all symbols slowly oscillating at infinity. Finally, we denote by the set of all symbols , for which (1.5) holds for all multi-indices and . The classes and were introduced by Grushin [11].

We denote by the set of all variable exponents for which there exist constants , , and a variable exponent such that

for almost all . Rabinovich and Samko observed in the proof of [23, Theorem 6.1] that if satisfies (1.1)–(1.3), then . It turns out that the class contains many interesting exponents which are not globally log-Hölder continuous (see [13]). In particular, there exists such that for every satisfying the function

belongs to .

As usual, we denote by the identity operator on a Banach space. Recall that a bounded linear operator on a Banach space is said to be Fredholm if there is an (also bounded linear) operator such that the operators and are compact. In that case the operator is called a regularizer for the operator .

Our second main result is the following sufficient condition for the Fredholmness of pseudodifferential operators on variable Lebesgue spaces.

Theorem 1.3.

Suppose and . If

(1.6)

then the operator is Fredholm on the variable Lebesgue space .

As it was the case with Theorem \@setrefth:boundedness, for satisfying (1.1)–(1.3) this result was established by Rabinovich and Samko [23, Theorem 6.1]. Notice that for such condition (1.6) is also necessary for the Fredholmness (see [23, Theorems 6.2 and 6.5]). Whether or not the necessity holds in the setting of Theorem \@setrefth:sufficiency, remains an open question.

The paper is organized as follows. In Section \@setrefsec:sharp, the Diening-Růžička generalization (see [10]) of the Fefferman-Stein sharp maximal theorem to the variable exponent setting is stated. Further, Diening’s results [8] on the duality and left-openness of the class are formulated. In Section \@setrefsec:pointwise we discuss a pointwise estimate relating the Fefferman-Stein sharp maximal operator of and for and . Such an estimate for the range of parameters , , and as in Theorem \@setrefth:boundedness was recently obtained by Michalowski, Rule, and Staubach [16]. Combining this key pointwise estimate with the sharp maximal theorem and taking into account that is bounded on for some whenever , we give the proof of Theorem \@setrefth:boundedness in Section \@setrefsec:proof.

Section \@setrefsec:Fredholmness is devoted to the proof of the sufficient condition for the Fredholmness of a pseudodifferentail operator with slowly oscillating symbol. In Section \@setrefsec:interpolation, we state analogues of the Riesz-Thorin and Krasnoselskii interpolation theorems for variable Lebesgue spaces. Section \@setrefsec:calculus contains the composition formula for pseudodifferential operators with slowly oscillating symbols and the compactness result for pseudodifferential operators with symbols in . Both results are essentially due to Grushin [11]. Section \@setrefsec:sufficiency contains the proof of Theorem \@setrefth:sufficiency. Its outline is as follows. From (1.6) it follows that there exist symbols and such that . Since , the operator is compact on all standard Lebesgue spaces. Its compactness on the variable Lebesgue space is proved by interpolation, since it is bounded on the variable Lebesgue space , where is the variable exponent from the definition of the class . Actually, the class is introduced exactly for the purpose to perform this step. Therefore is a right regularizer for on . In the same fashion it can be shown that is a left regularizer for . Thus is Fredholm.

\@xsect\@xsect

We start with the following simple but important property of variable Lebesgue spaces. Usually it is called the lattice property or the ideal property.

Lemma 2.1.

(see e.g. [9, Theorem 2.3.17]) Let be a measurable a.e. finite function. If , is a measurable function, and for a.e. , then and .

\@xsect

Let . For a cube , put

The Fefferman-Stein sharp maximal function is defined by

where the supremum is taken over all cubes containing .

It is obvious that is pointwise dominated by . Hence, by Lemma \@setrefle:lattice,

whenever . The converse is also true. For constant this fact goes back to Fefferman and Stein (see e.g. [25, Chap. IV, Section 2.2]). The variable exponent analogue of the Fefferman-Stein theorem was proved by Diening and Růžička [10].

Theorem 2.2.

(see [10, Theorem 3.6] or [9, Theorem 6.2.5]) If , then there exists a constant such that for all ,

\@xsect

Let . Given , the -th maximal operator is defined by

where the supremum is taken over all cubes containing . For this is the usual Hardy-Littlewood maximal operator. Diening [8] established the following deep duality and left-openness result for the class .

Theorem 2.3.

(see [8, Theorem 8.1] or [9, Theorem 5.7.2]) Let be a measurable function satisfying (1.1). The following statements are equivalent:

  1. is bounded on ;

  2. is bounded on ;

  3. there exists an such that is bounded on ;

  4. there exists a such that is bounded on .

\@xsect

One of the main steps in the proof of Theorem \@setrefth:boundedness is the following pointwise estimate.

Theorem 2.4.

(see [16, Theorem 3.3]) Let and with , , and . For every ,

where is a positive constant depending only on and the symbol .

This theorem generalizes the pointwise estimate by Miller [17, Theorem 2.8] for and by Álvarez and Hounie [1, Theorem 4.1] for with the parameters satisfying and .

Let . One of the main steps in the Rabinovich and Samko’s proof [23] of the boundedness on of the operator with is another pointwise estimate

for all , where is a positive constant independent of . It was proved in [23, Corollary 3.4] following the ideas of Álvarez and Pérez [2], where the same estimate is obtained for the Calderón-Zygmund singular integral operator in place of the pseudodifferential operator .

\@xsect

Suppose . Then, by Theorem \@setrefth:Diening, and there exists a number such that is bounded on . In other words, there exists a positive constant depending only on and such that for all ,

(2.1)

From Theorem \@setrefth:sharp it follows that there exists a constant such that for all ,

(2.2)

On the other hand, from Theorem \@setrefth:pointwise and Lemma \@setrefle:lattice we obtain that there exists a positive constant , depending only on and , such that

(2.3)

Combining (2.1)–(2.3), we arrive at

for all . It remains to recall that is dense in (see Lemma \@setrefle:density). ∎

\@xsect\@xsect

For a Banach space , let and denote the Banach algebra of all bounded linear operators and its ideal of all compact operators on , respectively.

Theorem 3.1.

Let , , be a.e. finite measurable functions, and let be defined for by

Suppose is a linear operator defined on .

  1. If for , then for all and

  2. If and , then for all .

Part (a) is proved in [9, Corollary 7.1.4] under the more general assumption that may take infinite values on sets of positive measure (and in the setting of arbitrary measure spaces). Part (b) was proved in [23, Proposition 2.2] under the additional assumptions that satisfy (1.1)–(1.3). It follows without these assumptions from a general interpolation theorem by Cobos, Kühn, and Schonbeck [4, Theorem 3.2] for the complex interpolation method for Banach lattices satisfying the Fatou property. Indeed, the complex interpolation space is isomorphic to the variable Lebesgue space (see [9, Theorem 7.1.2]), and have the Fatou property (see [9, p. 77]).

\@xsect

Let and be the class of all pseudodifferential operators with . By analogy with [11, Section 2] one can get the following composition formula (see also [21, Theorem 6.2.1] and [22, Chap. 4]).

Proposition 3.2.

If and , then their product belongs to and its symbol is given by

where .

Proposition 3.3.

Let . If , then .

Proof.

From Theorem \@setrefth:boundedness it follows that for all constant exponents . By [11, Theorem 3.2], . Hence, by the Krasnoselskii interpolation theorem (Theorem \@setrefth:interpolation(b) for constant with ), for all . ∎

\@xsect

The idea of the proof is borrowed from [11, Theorem 3.4] and [23, Theorem 6.1]. Let be such that if and if . For , put

From (1.6) it follows that there exists an such that

Then it is not difficult to check that

belongs to . It is also clear that .

From Proposition \@setrefpr:composition it follows that there exists a function such that

(3.1)

On the other hand, since

we have

(3.2)

Combining (3.1)–(3.2), we get

(3.3)

Since , there exist , , and such that

From Theorem \@setrefth:boundedness we conclude that all pseudodifferential operators considered above are bounded on , , and . Since , from Proposition \@setrefpr:compactness it follows that . Then, by Theorem \@setrefth:interpolation(b), . Therefore, from (3.3) it follows that is a right regularizer for . Analogously it can be shown that is also a left regularizer for . Thus is Fredholm on . ∎

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Departamento de Matemática
Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa
Quinta da Torre
2829–516 Caparica
Portugal

e-mail: oyk@fct.unl.pt

Department of Mathematics
College of William & Mary
Williamsburg, VA, 23187-8795
U.S.A.

e-mail: ilya@math.wm.edu

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