Hardy Type Operators In Local Vanishing Morrey Spaces On Fractal Sets

Abstract

Abstract We obtain two-weighted estimates for the Hardy type operators from local generalized Morrey spaces Lp,φ loc (X,w1) defined on an arbitrary underlying quasi-metric measure space (X, μ, ϱ) with the growth condition, to Lq,ψ loc(X,w2), where w1 = w1[ϱ(x, x0)], x0 ∈ X is a weight of radial type, while w2 = w2(x) may be an arbitrary weight. The proof allows to simultaneously treat a similar boundedness V Lp,φ loc (X,w1) → V Lq,ψ loc(X,w2) for vanishing Morrey spaces. We obtain sufficient conditions for such estimates in terms of some integral inequalities imposed on φ, ψ and w1.w2. We also specially treat the one weight case where w2(x) is also of radial type. We do not impose doubling condition on the measure μ, but base our result on the growth condition. The obtained results show the explicit dependence of the mapping properties of the Hardy type operators on the fractional dimension of the set (X, μ, ϱ). An application to spherical Hardy type operators is also given.