Work

Continuous Harmonic Functions on a Ball that are not in $$H^s$$ for $$s>1/2$$

Year: 2026

Type: article

Abstract: We show that there are harmonic functions on a ball $${\mathbb {B}_n}$$ B n of $$\mathbb {R}^n$$ R n , $$n\ge 2$$ n ≥ 2 , that are continuous, and even Hölder continuous, up to the boundary but not in the Sobolev space $$H^s(\mathbb {B}_n)$$ H s ( B n ) for s bigger than a certain sharp bound. The idea for the construction is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by Hadamard in 1906. To obtain examples in any dimension $$n\ge 2$$ n ≥ 2 we exploit certain series of spherical harmonics.

Source: Journal of Geometric Analysis

DOI: 10.1007/s12220-026-02429-6

Authors: Roberto Bramati, Matteo Dalla Riva, Brian Luczak

Institutions: University of Bergamo, Ghent University, University of Padua, DePaul University

Language: English

Topic: Advanced Harmonic Analysis Research

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,290