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Weighted Hardy-Type Inequalities in Variable Exponent Morrey-Type Spaces

DOI: 10.1155/2013/716029

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Abstract:

We study the boundedness of weighted multidimensional Hardy-type operators and of variable order , with radial weight , from a variable exponent locally generalized Morrey space to another . The exponents are assumed to satisfy the decay condition at the origin and infinity. We construct certain functions, defined by , , and , the belongness of which to the resulting space is sufficient for such a boundedness. Under additional assumptions on , this condition is also necessary. We also give the boundedness conditions in terms of Zygmund-type integral inequalities for the functions and . 1. Introduction Influenced by various applications, for instance, mechanics of the continuum medium and variational problems, in the last two decades the study of various mathematical problems in the spaces with nonstandard growth attracts the attention of researchers in various fields. This notion relates first of all to the generalized Lebesgue spaces , , known also as Lebesgue spaces with variable exponent . We refer to the existing books [1–3] in the field. This variable exponent boom naturally touched Morrey spaces. Morrey spaces (with constant exponents) in its classical version were introduced in [4] in relation to the study of partial differential equations and presented in various books; see, for example, [5–7]; we refer also to a recent overview of Morrey spaces in [8], where various generalizations of Morrey spaces may be also found. They were widely investigated during the last decades, including the study of classical operators of harmonic analysis, maximal, singular, and potential operators on Morrey spaces, and their generalizations were studied. We refer for instance to papers [9–16] and the references therein; in particular, Hardy operators in Morrey type spaces with constant were studied in [17–20]. The Morrey spaces with variable exponents and were introduced and studied in [21–23]. Generalized Morrey spaces ,?? with variable exponents were studied in [24]; see also another version of Morrey-type spaces in [25]; we also refer to [26] for the so-called complementary Morrey spaces of variable order in the spirit of ideas of [24]. In the above cited paper maximal, singular, and potential operators were studied. This paper seems to be the first one where Hardy-type integral inequalities are studied in Morrey-type spaces with variable exponents. Concerning Hardy-type inequalities and related problems and applications, we refer to the books [27, 28]. The paper is organized as follows. In Section 2, we give necessary preliminaries on variable exponent Lebesgue

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