Let be the class of functions bounded away from one and infinity and such that the Hardy-Littlewood maximal function is bounded on the variable Lebesgue space . We denote by the class of variable exponents for which with some , , and . Rabinovich and Samko  observed that each globally log-Hölder continuous exponent belongs to . We show that the class contains many interesting exponents beyond the class of globally log-Hölder continuous exponents.
On an Interesting Class of Variable Exponents] On an Interesting Class of Variable Exponents A. Yu. Karlovich]Alexei Yu. Karlovich
I. M. Spitkovsky]Ilya M. Spitkovsky
To Professor Stefan Samko on the occasion of his 70th birthday
Mathematics Subject Classification (2000). Primary 42B25; Secondary 46E30, 26A16.
Keywords. Variable Lebesgue space, variable exponent, globally log-Hölder continuous function, Hardy-Littlewood maximal operator..
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. We will always suppose that
Under these conditions, the space is separable and reflexive, and its dual is isomorphic to , where
(see e.g. [5, 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 .
and is constant outside some ball, then . Further, the behavior of at infinity was relaxed by D. Cruz-Uribe, A. Fiorenza, and C. Neugebauer [2, 3], who showed that if satisfies (1.2) and there exists a such that
The function is defined on the interval by
For and , put
and such that for some , , ,
Observe, however, that A. Lerner  (see also [5, Example 5.1.8]) constructed exponents discontinuous at zero or at infinity and such that, nevertheless, belong to . Thus neither (1.2) nor (1.3) is necessary for . For more informastion on the class we refer to [5, Chaps. 4–5].
Finally, we note that V. Kokilashvili and S. Samko [8, 9]; V. Kokilashvili, N. Samko, and S. Samko ; D. Cruz-Uribe, L. Diening, and P. Hästö  studied the boundedness of the Hardy-Littlewood maximal operator on variable Lebesgue spaces with weights under assumptions (1.1)–(1.3) or their analogues in the case of metric measure spaces.
We denote by the collection of all variable exponents for which there exist constants , , and a variable exponent such that
for almost all .
This class implicitly appeared in V. Rabinovich and S. Samko’s paper  (see also ). Its introduction is motivated by the fact that the boundedness of the Hardy-Littlewood maximal operator on implies the boundedness of many important linear operators on (see e.g. [5, Chap. 6]). If such a linear operator is also compact on the standard Lebesgue space , then, by a Krasnoselskii type interpolation theorem for variable Lebesgue spaces, it is compact on the variable Lebesgue space as well.
In [13, Theorem 5.1], the boundedness of the pseudodifferential operators with symbols in the Hörmander class on the variable Lebesgue spaces was established, provided that satisfies (1.1)–(1.3). Then the above interpolation argument was used in the proof of [13, Theorem 6.1] to study the Fredholmness of pseudodifferential operators with slowly oscillating symbols on . In particular, the following is implicitly contained in the proof of [13, Theorem 6.1].
Recently we generalized [13, Theorem 5.1] and proved that the pseudodifferential operators with symbols in the Hörmander class , where and , are bounded on variable Lebesgue spaces whenever (see [6, Theorem 1.2]). Further, [6, Theorem 1.3] delivers a sufficient condition for the Fredholmness of pseudodifferential operators with slowly oscillating symbols in the Hörmander class under the assumption that . The proof follows the same lines as V. Rabinovich and S. Samko’s proof of [13, Theorem 6.1] for exponents satisfying (1.1)–(1.3) and is based on the above mentioned interpolation argument.
The aim of this paper is to show that the class is much larger than the class of globally log-Hölder continuous exponents. Our first result says that all Nekvinda’s exponents belong to .
We have .
There exists a sufficiently small such that for every satisfying the function
belongs to .
The paper is organized as follows. For completeness, we give a proof of Theorem \@setrefth:RS in Section \@setrefsec:RS. Further, in Section \@setrefsec:Nekvinda we prove Theorem \@setrefth:Nekvinda. Section \@setrefsec:Lerner contains A. Lerner’s sufficient condition for in terms of mean oscillations of a function . In Section \@setrefsec:proof we show that Theorem \@setrefth:example follows from the results of Section \@setrefsec:Lerner.
In this subsection we give a proof of Theorem \@setrefth:RS. A part of this proof will be used in the proof of Theorem \@setrefth:Nekvinda in the next subsection.
Proof of Theorem \@setrefth:RS.
If we choose , then for ,
Hence and . If we choose such that , then and thus satisfies (1.1).
From (2.1) it follows that
Then, taking into account (2.2), we get
where is the constant from (1.3). Then
From (1.3) it follows that
Hence . Therefore
We will choose and subject to
Then, for ,
Therefore, for ,
where and are defined by (2.8) with in place of .
If for , then
that is, is monotone.
It is easy to see that for almost all ,
Taking into account (2.9), we obtain
and for ,
Therefore, there is a constant such that
Let . For a cube , put
We recall that the mean oscillation of over a cube is given by
If is a Lipschitz function with the Lipschitz constant and is a real-valued function, then for every ,
It is easy to see that
From this estimate we immediately get the statement. ∎
Given any cube , let
where is the center of .
Lemma 3.2 (see [10, Proposition 4.2]).
Theorem 3.3 (see [10, Theorem 1.2]).
There is a positive constant , depending only on , such that for any measurable function with
we have .
Let the function be as in Lemma \@setrefle:double-logarithm. Suppose and put
From Lemma \@setrefle:double-logarithm we know that
Since is a Lipschitz function with the Lipschitz constant equal to , we obtain from Lemma \@setrefle:Lipschitz-oscillation that
Fix and take the function such that
Since , we have for and then
If we take , then . Therefore . Hence
Thus is Lipschitz with the Lipschitz constant equal to .
Further, from (3.7) and Lemma \@setrefle:Lipschitz-oscillation we obtain
Let be the constant from Theorem \@setrefth:Lerner. Put
By Theorem \@setrefth:Lerner, and . Hence, (1.7) holds with , and . Thus . ∎
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Departamento de Matemática
Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa
Quinta da Torre
Department of Mathematics
College of William & Mary
Williamsburg, VA, 23187-8795