A condition equivalent to the Hölder continuity of harmonic functions on unbounded Lipschitz domains

Abstract

Our main result concerns the behavior of bounded harmonic functions on a domain in ${\mathbb{R}}^N$ which may be represented as a strict epigraph of a Lipschitz function on ${\mathbb{R}}^{N-1}$. Generally speaking, the result says that the Hölder continuity of a harmonic function on such a domain is equivalent to the uniform Hölder continuity along the straight lines determined by the vector ${\mathbf{e}}_N$, where ${\mathbf{e}}_1,{\mathbf{e}}_2,\dots ,{\mathbf{e}}_N$ is the base of standard vectors in ${\mathbb{R}}^N$. More precisely, let $\Psi$ be a Lipschitz function on ${\mathbb{R}}^{N-1}$, and $U$ be a real-valued bounded harmonic function on $E_{\Psi }=\{(x^{\prime },x_N):x^{\prime }\in {\mathbb{R}}^{N-1},x_N>\Psi (x^{\prime })\}$. We show that for $\alpha \in (0,1)$ the following two conditions on $U$ are equivalent: (a) There exists a constant $C$ such that \begin{equation*} | U(x',x_N) - U(x',y_N)|\le C |x_N - y_N|^α,\quad x'\in \mathbb {R}^{N-1}, x_N, y_N > Ψ(x'); \end{equation*} (b) There exists a constant $\tilde{C}$ such that \begin{equation*} |U(x) - U (y)|\le \tilde{C} |x-y|^α,\quad x, y\in E_Ψ. \end{equation*} Moreover, the constant $\tilde{C}$ depends linearly on $C$. The result holds as well for vector-valued harmonic functions and, therefore, for analytic mappings.

to appear in Communications in Contemporary Mathematics