A Necessary and Sufficient Condition for Hardy’s Operator in the Variable Lebesgue Space

Abstract

The variable exponent Hardy inequality , is proved assuming that the exponents , not rapidly oscilate near origin and . The main result is a necessary and sufficient condition on , generalizing known results on this inequality.

1. Introduction

There are a lot of examples in the theory of differential equations and analysis, in which the boundedness of the Hardy operator in some spaces is used essentially (a weighted Lebesgue space, Lorentz space, their weak spaces, and so on; see, e.g., [1–4]). The aim of present paper is to study a necessary and sufficient condition for the boundedness of Hardy operator in the weighted Lebesgue space with variable exponent . The investigation of variable exponent Lebesgue spaces is stimulated with the modeling of electrorhelogical fluids (see [5, 6]). That led to the development of regularity theory for the nonlinear elliptic and parabolic equations with partial derivatives (see, e.g., the bibliography in [7, 8]). Also, the required mathematical methods of analysis were elaborated to study the boundedness of principal integral operators (maximal operator, fractional operators, singular operator, commutators, and so on) in spaces (see the recent monographs [9, 10]).

In all probability, the first investigation of the variable exponent Hardy inequality was started in the works [11–15], subsequently in [16–20]. The variable exponent Hardy inequality was considered also in the recent works [21–25]. Since the Hardy operator is the simplest one among other integral operators, it seems logical to investigate the necessary and sufficient conditions for this operator in the first place. Though we have not so far succeeded in obtaining appropriate results in the general weighted space and in the case of general exponential functions, the representation of our results in presenting here form seems to us more attractive for comparision with the known results (see below).

More precisely, the subject of present paper is to study the norm inequality for the Hardy operator

Due to the cited above results, a necessary and sufficient condition for (1) take place if a regularity condition is assumed on at the origin. Namely, let , , , , both functions satisfy the condition then inequality (1) holds if and only if .

In our results, the exponent functions satisfy the following oscillation condition near origin:

This condition is weaker than known logarithmic condition (3); that is, (3) implies (4). We can give an example of exponential function for which condition (4) is satisfied but (3) fails: . Also using L’Hopital’s rule, it is not difficult to check that such a function satisfies the condition (5). Therefore, applying the assertion of our results below, we get new results on existence of inequality (1) (compare with the known results in [17] or [24]; the condition (3) was imposed there).

The following main result is obtained in this paper.

Theorem 1. Let and be measurable functions such that . Suppose there are limits and and the functions satisfy condition (4) near origin.
Then inequality (1) holds if and only if

2. Notation

As to the basic properties of spaces , we refer to [26, 27]. Throughout this paper, it is assumed that is a measurable function in taking its values from the interval with . The space of functions is introduced as the class of measurable functions in which has a finite modular (put also ). A norm in is given in the form (By or simply , we denote the norm over set .)

For , the space is a reflexive Banach space. The relation between the modular and norm is expressed by the following inequalities (see, e.g., [12]):

These inequalities allow us to perform our norm estimates in terms of a modular.

For the function ,   denotes the conjugate function of , and if . We denote by various positive constants whose values may vary at each appearance. We write if there exist positive constants such that . By , we denote the characteristic function of set .

We say a function is almost increasing (decreasing) if there exists a constant such that and for .

3. Proofs

Throughout the section, we assume that and are measurable functions such that , , and .

Lemma 2. Suppose is a measurable function such that , . and the condition (4) be satisfied. Then there exists a constant such that for any , , where the constant depends on , , and the constant from the condition (4).

Proof. Since satisfies the condition (4), it follows that, if , then and if then . By the same arguments,

Proof of Theorem 1. Consider the following.
Necessity. First, show that the function is almost decreasing if inequality (1) holds and the functions satisfy (4). In this way, we will show that where the constant does not depend on . For fixed , there exists an such that Let , , be such that . Insert a test function into inequality (1). Then and, therefore, . It follows from (1) that . This means that . Therefore, Hence, In the integral term, it easily follows from Lemma 2 and (4) for that Using this estimate and (17), (15), we see that respectively.
Using the Holder inequality for -norms, we see that since and by (19) Hence and then On the other hand, by Lemma 2 and (4) for , we have Using this inequality from (25), we get (11). Inequity (11) has been proved; that is, the function is almost decreasing.
Now, it follows from (11) and (20) that Hence,
Now, using (28), we will derive a Bari-Stechkin [28] type assertion in order to prove that the function is almost decreasing by some .
Indeed, put . Then, by (28), Integrating this inequality, Using (28), Now, it follows from Lemma 2 and (4) for that Therefore, that is, the function is almost decreasing by . This implies almost decreasing of the function by . Then it is easily seen that the condition (5) is satisfied.
This completes necessity of condition (5).
Sufficiency. Let the functions satisfy (4) and the condition (5). Show that (5) implies almost decreasing of by some . Put and repeat the arguments before.
We have Using (5), By Lemma 2 and (4) for , it follows that Hence, that is, is almost decreasing by .
Let be a measurable function such that . Then . We have to prove . By Minkowskii inequality for norms, where is a fixed number.
We will derive an estimate for every summand in (38). In this way, we will get a modular estimate for the corresponding terms in modular.
Denote By (37), for any , , where does not depend on . From (40) using , we get or
If , then, due to Holder’s inequality, for we have
Using (4) for and Lemma 2, it is not difficult to see the following estimate: where and , , with the constant not depending on .
Indeed, for and . Then to prove (44), it suffices to show , which is a simple consequence of (4) for , Lemma 2, and the fact that there exists a point such that .
For the second multilayer (43), we have the estimates Combining the estimates (43), (44), and (45) for and we have Now, taking into the account (42), here, we see that the last term is exceeded by On the other hand, using the assumptions and , Using (48) and , it follows from (47) that If , then we repeat all arguments with changed to . Indeed, it follows from the Holder inequality that By (4), for , we have (see similar arguments after (44)) Also, since the conditions , , and are assumed.
Hence, Therefore, by use of (42) for and , it follows that Let us note that since . Therefore, with .
It follows from (56) and (49) that, in both cases and , we have the same estimates (with different not depending on .). Denote again by the minimum of and and, taking into account (56) and (49), we get Due to Fubini’s theorem, this is exceeded by We have used that and the estimate which easily follows from .
Therefore, From this estimate and (38), it follows that It remains to get an estimate of far from origin. Since is separated from zero and infinity in , it suffices to note the estimate Here, the boundedness of first multiplier follows from the assumption. Boundedness of second multiplier follows from the condition and the assertion of Lemma 2.
Theorem 1 has been proved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the reviewer’s valuable comments that improved the paper. The first author is grateful to the Science Development Foundation under the President of Azerbaijan Republic (EIF-2012-2(6)-39/09/1). The authors are also grateful to the Scientific and Technological Research Council of Turkey for a scholarship (TÜBİTAK-BİDEB, 2012).