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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: WEIGHTED WEAK-TYPE INEQUALITIES FOR GENERALIZED HARDY OPERATORS | WEIGHTED WEAK-TYPE INEQUALITIES FOR GENERALIZED HARDY OPErAtORS A. L. BERNARDIS F. J. MARTÍN-REYES AND P. ORTEGA SALVADOR Received 13 June 2006 Accepted 21 September 2006 We characterize the pairs of weights v w for which the Hardy-Steklov-type operator Tf x g x Ịh x K x y f y dy applies Lp v into weak-Lq w q p assuming certain monotonicity conditions on g s h and K. Copyright 2006 A. L. Bernardis et al. 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. 1. Introduction Let us consider the Hardy-Steklov-type operator defined by h x Tf x g x 2 K x y f y dy f 0 1.1 s x where g is a nonnegative measurable function s and h are continuous and increasing functions x y s x s y h x h y defined on an interval a b such that s x h x for all x e a b and the kernel K x y defined on x y x e a b and s x y h x satisfies i K x y 0 ii it is increasing and continuous in x and decreasing in y iii K x z D K x h y K y z for y x and s x z h y where the constant D 1 is independent of x y and z. Gogatishvili and Lang 3 characterized the pairs of weights for the strong- and weak-type p q inequalities for the operator T in the case p q. Actually in 3 the authors deal with Banach functions spaces with some extra condition. On the other hand Chen and Sinnamon 2 have characterized the weighted strong-type inequality for 1 p q 00 in terms of a normalizing measure. In both papers they work with more general functions s h and K. Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 62426 Pages 1-10 DOI 10.1155 JIA 2006 62426 2 Generalized Hardy operators The goal of this paper is to characterize the weighted weak-type inequalities in the case q p. It is well known that strong-type inequalities for the operator T can be deduced directly from the corresponding ones for g x 1 but this is not the case .
