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

摘要

We study thep·→q·boundedness of weighted multidimensional Hardy-type operatorsHwα·andℋwα·of variable orderαx, with radial weightwx, from a variable exponent locally generalizedMorrey spaceℒp·,φ·ℝn,wto anotherℒq·,ψ·ℝn,w. The exponents are assumed to satisfy the decay condition atthe origin and infinity. We construct certain functions, defined byp,α, andφ, the belongness of which to theresulting spaceℒq·,ψ·ℝn,wis sufficient for such a boundedness. Under additional assumptions onφ/w, this condition is also necessary. We also give the boundedness conditions in terms of Zygmund-type integral inequalities for the functionsφandφ/w.