Lipschitz type Continuity for Solutions of the Invariant Laplacian Poisson Equations

Abstract

The main aim of this paper is to investigate the Lipschitz type continuity for the solutions of the invariant Laplacian Poisson equation Δαu(x)=ψ(x) in Bn, where u|Sn-1=ϕ∈L∞Sn-1,Rn, ψ∈L∞(Bn,Rn) and Δα is the invariant Laplacian operator in Rn for n≥3. In order to reach this goal, firstly, we show that if u∈C2Bn,Rn∩CBn¯,Rn is a solution to the above equation and ψ(x)1-|x|2-1 is integrable in Bn, then u=Pα[ϕ]-Gα[ψ], where Pα[ϕ] and Gα[ψ] denote the Poisson integral of ϕ and Green integral of ψ with respect to Δα, respectively. Secondly, we prove that if u=Pα[ϕ]-Gα[ψ]∈C2Bn,Rn∩CBn¯,Rn and ψ(x)1-|x|2-1 is integrable in Bn, then u is a solution to the above equation. Thirdly, we prove the main result of this paper. We show that if 0<β≤1, α<1-β, u=Pα[ϕ]-Gα[ψ]∈C2Bn,Rn∩CBn¯,Rn, and if there are two non-negative constants L, M such that |ϕ(ξ)-ϕ(η)|≤L|ξ-η|β for all ξ,η∈Sn-1 and |ψ(x)|≤M(1-|x|2)β for all x∈Bn, then there exists a positive constant N such that for any |u(x)-u(y)|≤N|x-y|β in Bn¯. Finally, we consider the local Lipschitz type continuity and we obtain a local spatial version of Privalov theorem for α-harmonic mappings.