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

Abstract

We study the p·→q· boundedness of weighted multidimensional Hardy-type operators Hwα· and ℋwα· of variable order αx, with radial weight wx, from a variable exponent locally generalized Morrey space ℒp·,φ·ℝn,w to another ℒq·,ψ·ℝn,w. The exponents are assumed to satisfy the decay condition at the origin and infinity. We construct certain functions, defined by p, α, and φ, the belongness of which to the resulting space ℒq·,ψ·ℝn,w is sufficient for such a boundedness. Under additional assumptions on φ/w, this condition is also necessary. We also give the boundedness conditions in terms of Zygmund-type integral inequalities for the functions φ and φ/w.