Uniqueness theorems for weighted harmonic functions in the upper half-plane

TL;DR

This work analyzes the uniqueness of $\alpha$-harmonic functions in the open upper half-plane under vanishing boundary data and growth conditions. It develops a general obstruction framework via the polynomials $p_{k,\alpha}$, yielding a representation $u(z)=\sum_{k=0}^n c_k(\Im z)^{\alpha+1} p_{k,\alpha}(z)$ for functions vanishing on $\mathbb{R}$, and shows a strong uniqueness property for $\alpha\neq0$ under a mild infinity-vanishing condition, with distributional variants. For the classical case $\alpha=0$, the paper proves sharp uniqueness results along geodesics (two distinct geodesics are necessary and sufficient) and along rays, formulated through the novel concept of admissible functions of angles, including explicit constructions and minimality results. The approach blends Poisson integral representations, angular derivatives of Poisson kernels, a disk-model reduction via a Möbius map, and arithmetic geometry of angles, linking hypergeometric function techniques with geometric function theory to illuminate boundary uniqueness phenomena in weighted harmonic settings.

Abstract

We consider a class of weighted harmonic functions in the open upper half-plane known as $α$-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case ($α\neq0$) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case ($α=0$) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. Our findings shed new light on the theory of harmonic functions, for which we provide uniqueness results under vanishing conditions at infinity along a) geodesics, and b) rays emanating from the origin. The geodesic uniqueness results require vanishing on two distinct geodesics which is best possible. The ray uniqueness results involves an arithmetic condition which we analyze by introducing the concept of an admissible function of angles. We show that the arithmetic condition is to the point and that the set of admissible functions of angles is minimal with respect to a natural partial order.

Theorems & Definitions (100)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Proposition 3.1
• proof