A condition equivalent to the H\"{o}lder 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 H\"{o}lder continuity of a harmonic function on such a domain is equivalent to the uniform H\"{o}lder 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 $E_\Psi=\{(x',x_N): x'\in\mathbb{R}^{N-1}, x_N>\Psi(x')\}$. 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|^\alpha,\quad x'\in \mathbb {R}^{N-1}, x_N, y_N > \Psi (x'); \end{equation*} (b) There exists a constant $\tilde {C}$ such that \begin{equation*} |U(x) - U (y)|\le \tilde{C} |x-y|^\alpha,\quad x, y\in E_\Psi. \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.