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

TL;DR

The paper addresses Lipschitz (Hölder) regularity for bounded harmonic functions on the upper half-space $\mathbb{R}^n_+$, establishing how boundary Lipschitz control of $|U(x',0)|$ propagates to interior Hölder continuity. The authors derive explicit gradient bounds in terms of $x_n^{\alpha-1}$ using the Poisson kernel representation and Khavinson–Maz'ya type estimates, and deduce uniform Hölder estimates for $|U(x)-U(y)|$ in the domain. They prove the equivalence of boundary and interior Lipschitz semi-norms $\|U\|_1,\|U\|_2,\|U\|_3$, and extend Pavlović-type results to the upper half-space through a sequence of lemmas and sharp constants. These results provide a precise boundary-to-interior Lipschitz transfer for harmonic functions in a noncompact domain and add to the Khavinson problem literature with explicit constants.

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 $α\in (0,1)$ there exists a constant $C$ such that for every $x'\in \mathbb{R}^{n-1}$ we have $| |U|(x',x_n) - |U|(x',0)|\le Cx_n^α,\, x_n\in (0,\infty)$. Then there exists a constant $\tilde {C}$ such that $|U(x) - U (y)| \le \tilde{C} |x-y|^α,\, x,y\in \mathbb{R}^{n}_+$.

Theorems & Definitions (19)

• Proposition 1.1: Pavlović
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1: Kresin-Maz'ya
• Proposition 2.2: Liu
• Proposition 2.3: the Schwarz lemma for harmonic functions
• Lemma 2.4
• proof