International Journal of Analysis

Volume 2014 (2014), Article ID 606012, 5 pages

http://dx.doi.org/10.1155/2014/606012

Research Article

On the Logarithmic Regularity Conditions for the Variable Exponent Hardy Type Inequality

Aziz Harman and Mustafa Özgür Keleş

Faculty of Education, Dicle University, 21280 Diyarbakır, Turkey

Received 11 November 2013; Revised 24 January 2014; Accepted 10 February 2014; Published 23 April 2014

Academic Editor: Yongsheng S. Han

Copyright © 2014 Aziz Harman and Mustafa Özgür Keleş. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We discuss a logarithmic regularity condition in a neighborhood of the origin and infinity on the exponent functions and for the variable exponent Hardy inequality to hold.

1. Introduction

There are several papers devoted to the variable exponent Hardy inequality:

Let be positive measurable functions and , ; then inequality (1) and (2) hold under certain conditions on the weight functions , and the exponents , . Two types of conditions arise here: a balance condition on the weights and a regularity condition on the exponents (see below). Necessary and sufficient conditions for the validity of general inequality (1) were found in [1] for the case of , in [2] for cases and , in [3] for cases and , and in [4] for mixed cases and   ( and ). Some special cases of (1) are studied in [5–10] too.

So, the inequality

is a particular case of (1) when , and , where is the Hardy operator. For the constant exponents , this inequality holds if , (see, e.g., [11]). Necessary and sufficient conditions on the , for inequality (2) to hold are and if the exponents , are continuous near the origin and infinity such that the conditions are satisfied (see, e.g., [5–7, 9]). In [8] (see also [10]) it was proved that the condition is necessary for inequality (2) to hold if one of these exponents is a constant. Also, it was proved in [10] that the condition is sufficient for inequality (2) to hold on bounded interval if a constant in the condition for the satisfies and the is zero. The condition is weaker then . Also the function satisfies the condition but does not satisfy .

In this note, we will focus on the results of sufficiency and necessity of regularity conditions , , and below for inequality (2) to hold.

The space of functions is introduced as the class of measurable functions in , which have a finite modular. A norm in is given in the form

As to the basic properties of spaces , we refer to [12].

2. Main Results

We will state some sufficiency and necessity assertions concerning inequality (2). Along the way, it will be given a proof for two elementary estimates that we had used.

Let us introduce the following classes of measurable functions. We say, is in the class if

is in the class if

and is in the class if

Theorem 1 (see [8]). Suppose and is an increasing function on such that is continuous at and , ; then for the inequality

to hold it is necessary that .

Theorem 2 (see [8]). Suppose and is a decreasing function on such that is continuous at and , ; then for the inequality

to hold it is necessary that .

Theorem 3. Suppose is measurable functions such that , , , ; then for the inequality

to hold it is sufficient that , whenever and .

Theorem 4. Let , and let . There exists a sequence and a function , satisfying the conditions , and

or

violating inequality (2).

Theorem 5. Let , and let . Then there exists a sequence and a function satisfying the conditions , , and

or

violating inequality (2).

3. Proof of Main Results

For the proof of Theorems 1 and 2 we refer to [8]. Other proofs of these theorems are given in [10]. The proof of Theorem 3 also is given in [8]. Here we derive an alternative proof of that theorem using the general results of [2, 4].

In the proof of main results we use the following elementary Lemma.

Lemma 6. Suppose is a measurable function such that and , ; then it holds the estimate

for and the estimate

for .

Proof of Theorem 3. To prove Theorem 3 we apply the results of [2, 4], where it was proved that the following conditions

are necessary and sufficient for inequality (1) to hold if , , and the regularity conditions are satisfied:

where , .

In Theorem 3, we have accepted that and , . It is easy to show that the conditions imply . Therefore, it follows from Lemma 6 that

for by some . Also the conditions imply . Therefore, it follows from Lemma 6 that

for by some . Integrating these inequalities over the intervals and , respectively, we get

To complete the proof of Theorem 3 it suffices to apply estimates (20) to verify conditions (16) and (17). Now, Theorem 3 follows from the upper refereed results of the works [2, 4].

Proof of Lemma 6. Let and let ; then we have , where is a certain number from the interval . If and then by condition we have

Therefore, for we have the estimation

where the positive constant depends on , , , . Same inequality holds for the function . Indeed, for and we have . If and by condition we have

By using these inequalities and by the representation

we have estimate (14).

To show estimate (15) note that for and we have , where the is a certain number. If and then by the condition we have

Combining the estimates for the functions , for by the presentation

we get estimate (15). To complete the proof of Lemma 6, note that the condition is equivalent to

and the is equivalent to

respectively.

Proof of Theorem 4. Let us assume that ; . Fix . We define the step function

Here is a sequence of positive numbers that satisfies the condition as . Then as and condition (10) is fulfilled for the function . We have

as . The last relation shows violating of inequality (2) for sufficiently large .

We define for . Fix . We also define the step function

where as . We have as and condition (11) holds for the function . Furthermore,

as , which contradicts (2) for sufficiently large .

Proof of Theorem 5. Let us assume that ; . Fix . We define the step function as

where . Then,

as . The last relation contradicts the validity of inequality (2).

We define ; . Fix , where the function is defined as

where . Then,

as which contradicts inequality (2).

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the anonymous referee for carefully reading the paper and making useful comments.

References

F. I. Mamedov and A. Harman, “On a weighted inequality of Hardy type in spaces ,” Journal of Mathematical Analysis and Applications, vol. 353, no. 2, pp. 521–530, 2009.

F. I. Mamedov and Y. Zeren, “On equivalent conditions for the general weighted Hardy type inequality in space ,” Zeitschrift für Analysis und ihre Anwendungen, vol. 31, no. 1, pp. 55–74, 2012.

F. I. Mamedov and A. Harman, “On a Hardy type general weighted inequality in spaces ,” Integral Equations and Operator Theory, vol. 66, no. 4, pp. 565–592, 2010.

D. Cruz-Uribe and F. I. Mamedov, “On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces,” Revista Matemática Complutense, vol. 25, no. 2, pp. 335–367, 2012.

L. Diening and S. Samko, “Hardy inequality in variable exponent Lebesgue spaces,” Fractional Calculus & Applied Analysis, vol. 10, no. 1, pp. 1–18, 2007.

P. Harjulehto, P. Hästö, and M. Koskenoja, “Hardy's inequality in a variable exponent Sobolev space,” Georgian Mathematical Journal, vol. 12, no. 3, pp. 431–442, 2005.

A. Harman and F. I. Mamedov, “On boundedness of weighted Hardy operator in and regularity condition,” Journal of Inequalities and Applications, vol. 2010, Article ID 837951, 14 pages, 2010.

A. Harman, “On necessary condition for the variable exponent Hardy inequality,” Journal of Function Spaces and Applications, vol. 2012, Article ID 385925, 6 pages, 2012.

R. Mashiyev, B. Çekiç, F. I. Mamedov, and S. Ogras, “Hardy's inequality in power-type weighted spaces,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 289–298, 2007.

F. I. Mamedov, “On Hardy type inequality in variable exponent Lebesgue space ,” Azerbaijan Journal of Mathematics, vol. 2, no. 1, pp. 90–362, 2012.

A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type, World Scientific, River Edge, NJ, USA, 2003.

O. Kováčik and J. Rákosník, “On spaces and ,” Czechoslovak Mathematical Journal, vol. 41, no. 116, pp. 592–618, 1991.