Two-weight boundedness for local fractional maximal and applications

Abstract

Given Ω a proper open subset of a metric space with the weak homogeneity property and a measure μ doubling on certain local balls, we give sufficient conditions about local weights for the two-weight boundedness of the local fractional maximal operator acting on weighted Lebesgue spaces. As applications we obtain analogous results for singular and fractional type operators and their commutators. As a further application we present an a priori estimate for solutions of Δ mu= f in Ω , acting in weighted Sobolev spaces involving the distance to the boundary and different local weights.