Abstract
Two-weighted norm estimates with general weights for Hardy-type transforms and potentials in variable exponent Lebesgue spaces defined on quasimetric measure spaces are established. In particular, we derive integral-type easily verifiable sufficient conditions governing two-weight inequalities for these operators. If exponents of Lebesgue spaces are constants, then most of the derived conditions are simultaneously necessary and sufficient for corresponding inequalities. Appropriate examples of weights are also given.
Highlights
Two-weighted norm estimates with general weights for Hardy-type transforms and potentials in variable exponent Lebesgue spaces defined on quasimetric measure spaces X, d, μ are established
We study the two-weight problem for Hardy-type and potential operators in Lebesgue spaces with nonstandard growth defined on quasimetric measure spaces X, d, μ
The conditions are simultaneously necessary and sufficient for corresponding Journal of Inequalities and Applications inequalities when the weights are of special type and the exponent p of the space is constant
Summary
We study the two-weight problem for Hardy-type and potential operators in Lebesgue spaces with nonstandard growth defined on quasimetric measure spaces X, d, μ. Weighted inequalities for classical operators in Lpw· spaces, where w is a power-type weight, were established in the papers 10–12, 15–19 , while the same problems with general weights for Hardy, maximal, and fractional integral operators were studied in 10, 20–25. It should be emphasized that in the classical Lebesgue spaces the two-weight problem for fractional integrals is already solved see 26, 27 , but it is often useful to construct concrete examples of weights from transparent and verifiable conditions. The paper is organized as follows: in Section 1, we give some definitions and prove auxiliary results regarding quasimetric measure spaces and the variable exponent Lebesgue spaces; Section 2 is devoted to the sufficient governing two-weight inequalities for Hardytype operators defined on quasimetric measure spaces, while in Section 3 we study the twoweight problem for potentials defined on X. Throughout the paper is denoted the function p x / p x − 1 by the symbol p x
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