The main aim of this paper is to investigate the Lipschitz type continuity for the solutions of the invariant Laplacian Poisson equation \(\Delta _{\alpha } u(x)=\psi (x)\) in \({\mathbb {B}}^{n}\) , where \(u|_{{\mathbb {S}}^{n-1}}=\phi \in L^{\infty }\left( {\mathbb {S}}^{n-1}, {\mathbb {R}}^{n}\right) \) , \(\psi \in L^{\infty }({\mathbb {B}}^{n},{\mathbb {R}}^{n})\) and \(\Delta _{\alpha } \) is the invariant Laplacian operator in \({\mathbb {R}}^{n}\) for \(n\ge 3\) . In order to reach this goal, firstly, we show that if \(u \in C^{2}\left( {\mathbb {B}}^{n}, {\mathbb {R}}^{n}\right) \cap C\left( \overline{{\mathbb {B}}^{n}},{\mathbb {R}}^{n}\right) \) is a solution to the above equation and \( \psi (x) \left( 1-|x|^{2}\right) ^{-1}\) is integrable in \({\mathbb {B}}^{n}\) , then \(u=P_{\alpha }[\phi ]-G_{\alpha }[\psi ]\) , where \(P_{\alpha }[\phi ]\) and \(G_{\alpha }[\psi ]\) denote the Poisson integral of \( \phi \) and Green integral of \( \psi \) with respect to \(\Delta _{\alpha }\) , respectively. Secondly, we prove that if \(u=P_{\alpha }[\phi ]-G_{\alpha }[\psi ] \in C^{2}\left( {\mathbb {B}}^{n}, {\mathbb {R}}^{n}\right) \cap C\left( \overline{{\mathbb {B}}^{n}},{\mathbb {R}}^{n}\right) \) and \( \psi (x) \left( 1-|x|^{2}\right) ^{-1}\) is integrable in \({\mathbb {B}}^{n}\) , then u is a solution to the above equation. Thirdly, we prove the main result of this paper. We show that if \(0<\beta \le 1\) , \(\alpha <1-\beta \) , \(u=P_{\alpha }[\phi ]-G_{\alpha }[\psi ] \in C^{2}\left( {\mathbb {B}}^{n}, {\mathbb {R}}^{n}\right) \cap C\left( \overline{{\mathbb {B}}^{n}},{\mathbb {R}}^{n}\right) \) , and if there are two non-negative constants L, M such that \(|\phi (\xi ) - \phi (\eta )| \le L |\xi - \eta |^{\beta }\) for all \(\xi ,\eta \in {\mathbb {S}}^{n-1}\) and \(|\psi (x)| \le M (1 - |x|^2)^{\beta }\) for all \(x\in {\mathbb {B}}^n\) , then there exists a positive constant N such that for any \(|u(x) - u(y)| \le N |x - y|^{\beta } \) in \(\overline{{\mathbb {B}}^n}\) . Finally, we consider the local Lipschitz type continuity and we obtain a local spatial version of Privalov theorem for \(\alpha \) -harmonic mappings.
