%0 Journal Article
%T On the Logarithmic Regularity Conditions for the Variable Exponent Hardy Type Inequality
%A Aziz Harman
%A Mustafa £¿zg¨¹r Kele£¿
%J International Journal of Analysis
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/606012
%X 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¨C10] 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¨C7, 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
%U http://www.hindawi.com/journals/ijanal/2014/606012/