# [

###### Abstract

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 [13] 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

(1.1) |

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 .

Assume that (1.1) is fulfilled. L. Diening [4] proved that if satisfies

(1.2) |

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

(1.3) |

then . Following [5, Section 4.1], we will say that if conditions (1.2)–(1.3) are fulfilled, then is globally log-Hölder continuous.

A. Nekvinda [11, 12] relaxed condition (1.3). To formulate his results, we will need the notion of iterated logarithms. Put

The function is defined on the interval by

For and , put

We say that a measurable function belongs to the Nekvinda class if conditions (1.1)–(1.2) are fulfilled and there exists a monotone function satisfying

(1.4) |

and such that for some , , ,

(1.5) |

and

(1.6) |

for some . According to [12, Theorem 2.2], . In particular, all locally log-Hölder continuous (that is, satisfying (1.2)) radially monotone exponents with monotone satisfying (1.4)–(1.5) belong to .

Observe, however, that A. Lerner [10] (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 [7]; D. Cruz-Uribe, L. Diening, and P. Hästö [1] 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

(1.7) |

for almost all .

This class implicitly appeared in V. Rabinovich and S. Samko’s paper [13] (see also [14]). 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 .

###### Theorem 1.2.

We have .

Modifying A. Lerner’s example [10], we further prove that there are exponents in that do not satisfy (1.3).

###### Theorem 1.3.

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.

Suppose satisfies (1.1)–(1.3). Let , , and be such that (1.7) holds. Then

(2.1) |

If we choose , then for ,

(2.2) |

Therefore

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

(2.3) |

Now put

where is the constant from (1.3). Then

(2.4) |

From (1.3) it follows that

Hence . Therefore

(2.5) |

From (2.4) and (2.5) we obtain

(2.6) |

From estimates (2.3), (2.6) and (1.2), (1.3) for the exponent we obtain that the exponent satisfies (1.2) and (1.3). Therefore and thus belongs to . ∎

Suppose . Let , , and be such that (1.7) holds. In the previous subsection we proved that if and , then satisfies (1.1)–(1.2).

Since , there exists a monotone function such that (1.4)–(1.6) are fulfilled. Let

(2.7) |

Put

(2.8) |

We will choose and subject to

Then, for ,

(2.9) |

Therefore, for ,

and

(2.10) |

where and are defined by (2.8) with in place of .

If for , then

Thus

that is, is monotone.

It is easy to see that for almost all ,

Taking into account (2.9), we obtain

(2.11) |

and for ,

From this equality and inequalities (2.2) and (2.9) we get for ,

Therefore, there is a constant such that

(2.12) |

Since satisfies (1.4)–(1.6), from (2.10)–(2.12) it follows that satisfies (1.4)–(1.6), too. Thus . By [12, Theorem 2.2], , which finishes the proof of . ∎

Let . For a cube , put

We recall that the mean oscillation of over a cube is given by

###### Lemma 3.1.

If is a Lipschitz function with the Lipschitz constant and is a real-valued function, then for every ,

###### Proof.

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]).

If

then

###### 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

and

Then

(3.1) |

From Lemma \@setrefle:double-logarithm we know that

(3.2) |

Since is a Lipschitz function with the Lipschitz constant equal to , we obtain from Lemma \@setrefle:Lipschitz-oscillation that

(3.3) |

From (3.1), (3.3), and it follows that

(3.4) |

Fix and take the function such that

Then

Therefore

and

Since , we have for and then

(3.5) |

Hence

If we take , then . Therefore . Hence

(3.6) |

and

(3.7) |

Thus is Lipschitz with the Lipschitz constant equal to .

Put for . Then from (3.5)–(3.6) it follows that

(3.8) |

Further, from (3.7) and Lemma \@setrefle:Lipschitz-oscillation we obtain

(3.9) |

From (3.8)–(3.9) and we deduce

(3.10) |

Let be the constant from Theorem \@setrefth:Lerner. Put

If , then from (3.1), (3.4) and (3.8), (3.10) it follows that , , and

By Theorem \@setrefth:Lerner, and . Hence, (1.7) holds with , and . Thus . ∎

## References

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Departamento de Matemática

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