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

• Published in: Journal d'analyse mathématique (Jerusalem) Vol. 152; no. 1; pp. 317 - 359
• Main Authors: Olofsson, Anders, Wittsten, Jens
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 01.04.2024; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Analysis; Dirichlet problem; Dynamical Systems and Ergodic Theory; Functional Analysis; Geodesy; Half planes; Harmonic functions; Infinity; Matematik; Matematisk analys; Mathematical Analysis; Mathematical Sciences; Mathematics; Mathematics and Statistics; Natural Sciences; Naturvetenskap; Partial Differential Equations; Polynomials; Uniqueness theorems
• ISSN: 0021-7670; 1565-8538; 1565-8538
• DOI: 10.1007/s11854-023-0298-8

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 ≠ 0) 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. These findings shed new light on the theory of harmonic functions, for which we provide sharp uniqueness results under vanishing conditions at infinity along geodesics or along rays emanating from the origin.