On potentials in generalized Hölder spaces over uniform domains in ℝ n

Abstract

We show that Riesz-type potential operators of order α over uniform domains Ω in ℝn map the subspace \(H_{0}^{\lambda }(\Omega )\) of functions in Holder space Hλ(Ω) vanishing on ∂Ω, into the space Hλ+α(Ω), if λ+α≤1. This is proved in a more general setting of generalized Holder spaces with a given dominant of continuity modulus. Statements of such a kind are known for instance for the whole space ℝn or more generally for metric measure spaces with cancellation property. In the case of domains in ℝn when the cancellation property fails, our proofs are based on a special treatment of potential of a constant function.