Abstract
Abstract A considerable number of research has been carried out on the generalized Lebesgue spaces L p(x) and boundedness of different integral operators therein. In this study, a new approach for weighted increasing near the origin and decreasing near infinity exponent function that provides a boundedness of the Hardy’s operator in variable exponent space is given.
Highlights
Variable exponent studies have been stimulated by problems of elasticity, fluid dynamics, calculus of variations and differential equations with a non-standard growth condition, we refer to monographs [7, 11, 12, 13, 15, 18, 21, 23, 29, 30, 31]
The boundedness problems for weighted Hardy’s operator in variable exponent Lebesgue spaces Lp(.) are well studied, we refer to monographs [4, 9, 14, 16, 17, 19, 20, 22, 24, 26, 27, 28]
A necessary and sufficient condition that assumes a log-regularity of exponent function near origin and infinity has been proved in [1, 5, 6, 8, 10, 17, 23]
Summary
Variable exponent studies have been stimulated by problems of elasticity, fluid dynamics, calculus of variations and differential equations with a non-standard growth condition, we refer to monographs [7, 11, 12, 13, 15, 18, 21, 23, 29, 30, 31]. The boundedness problems for weighted Hardy’s operator in variable exponent Lebesgue spaces Lp(.) are well studied, we refer to monographs [4, 9, 14, 16, 17, 19, 20, 22, 24, 26, 27, 28] In this connection, a necessary and sufficient condition that assumes a log-regularity of exponent function near origin and infinity has been proved in [1, 5, 6, 8, 10, 17, 23]. The weight functions v and w are assumed to be measurable and having non-negative finite values almost everywhere in (0, ∞)
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