Lipschitz conditions on bounded harmonic functions on the upper half-space

TL;DR

This paper investigates Lipschitz conditions for bounded harmonic functions on the upper half-space, establishing that certain boundary regularity implies interior Hölder continuity of the functions.

Contribution

It proves that boundary Hölder continuity of bounded harmonic functions on the upper half-space implies their interior Hölder continuity with a related exponent.

Findings

Boundary Hölder condition implies interior Hölder continuity.

Established explicit relation between boundary regularity and interior smoothness.

Results extend understanding of harmonic function regularity in half-spaces.

Abstract

This work is devoted to Lipschitz conditions on bounded harmonic functions on the upper half-space in $\mathbb {R}^n$. Among other results we prove the following one. Let $U(x',x_n)$ be a real-valued bounded harmonic function on the upper half-space $\mathbb {R}^n_+ = \{(x',x_n):x'\in \mathbb{R}^{n-1}, x_n\in (0,\infty)\}$, which is continuous on the closure of this domain. Assume that for $\alpha\in (0,1)$ there exists a constant $C$ such that for every $x'\in \mathbb{R}^{n-1}$ we have $| |U|(x',x_n) - |U|(x',0)|\le Cx_n^\alpha,\, x_n\in (0,\infty)$. Then there exists a constant $\tilde {C}$ such that $|U(x) - U (y)| \le \tilde{C} |x-y|^\alpha,\, x,y\in \mathbb{R}^{n}_+$.