Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces

Abstract

We study commutators of weighted fractional Hardy-type operators within the frameworks of local generalized Morrey spaces over quasi-metric measure spaces for a certain class of “radial” weights. Quasi-metric measure spaces may include, in particular, sets of fractional dimentsions. We prove theorems on the boundedness of commutators with CMO coefficients of these operators. Given a domain Morrey space L^p,ϕ(X) for the fractional Hardy operators or their commutators, we pay a special attention to the study of the range of the exponent q of the target space L^q,ψ(X). In particular, in the case of classical Morrey spaces, we provide the upper bound of this range which is greater than the known Adams exponent. MSC 2010: Primary 46E30; Secondary 42B35, 42B25, 47B38 Key Words and Phrases: Morrey space; weighted fractional Hardy operators; commutators; BMO; CMO; quasi-metric measure spaces; growth condition; homogeneous spaces; quasi-monotone weights