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.
Categories: math.CV