Local Fractional and singular integrals on open subsets

Abstract

For a proper open set Ω 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 Ω, we introduce local operators of singular and fractional type and study their boundedness properties on weighted Lp(Ω), 1 ≤ p < ∞, for weights in local Muckenhoupt classes.