Local Maximal Function and Weights in a General Setting

Abstract

For a proper open set "omega" immersed in a metric space with the weak homogeneity property, and given a measure μ doubling on a certain family of balls lying “well inside” of "omega", we introduce a local maximal function and characterize the weights w for which it is bounded on L p("omega", wdμ) when 1 < p < ∞ and of weak type (1, 1). We generalize previous known results and we also present an application to interior Sobolev’s type estimates for appropriate solutions of the differential equation "delta" mu = f , satisfied in an open proper subset "omega" of Rn. Here, the data f belongs to some weighted L p space that could allow functions to increase polynomially when approaching the boundary of "omega".