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Weighted Hardy Operators in Complementary Morrey Spaces

DOI: 10.1155/2012/283285

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We study the weighted -boundedness of the multidimensional weighted Hardy-type operators and with radial type weight , in the generalized complementary Morrey spaces defined by an almost increasing function . We prove a theorem which provides conditions, in terms of some integral inequalities imposed on and , for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the function and the weight are power functions. We also prove that the spaces over bounded domains ? are embedded between weighted Lebesgue space with the weight and such a space with the weight , perturbed by a logarithmic factor. Both the embeddings are sharp. 1. Introduction Hardy operators and related Hardy inequalities are widely studied in various function spaces, and we refer to the books [1–4] and references therein. They continue to attract attention of researchers both as an interesting mathematical object and a useful tool for many purposes: see for instance the recent papers [5, 6]. Results on weighted estimations of Hardy operators in Lebesgue spaces may be found in the abovementioned books. In the papers [7–9] the weighted boundedness of the Hardy type operators was studied in Morrey spaces. In this paper we study multi-dimensional weighted Hardy operators where , in the so called complementary Morrey spaces. The one-dimensional case will include the versions adjusted for the half-axis , so that in the sequel with may be read either as or . The classical Morrey spaces , defined by the norm where , are well known, in particular, because of their usage in the study of regularity properties of solutions to PDE; see for instance the books [10–12] and references therein. There are also known various generalizations of the classical Morrey spaces , and we refer for instance to the surveying paper [13]. One of the direct generalizations is obtained by replacing in (1.3) by a function , usually satisfying some monotonicity type conditions. We also denote it as without danger of confusion. Such spaces appeared in [14, 15] and were widely studied in [16, 17]. The spaces , defined by the norm where , are known as generalized local Morrey spaces Hardy type operators (1.1) in the spaces have been studied in [7–9]. The norm in Morrey spaces controls the smallness of the integral over small balls (and also a possible growth of this integral for in the case is unbounded). There are also known spaces , called complementary Morrey spaces, with the norm controlling possible growth, as , of the integral over exterior of balls. Such


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