Continuous Harmonic Functions on a Ball that are not in $$H^s$$ for $$s>1/2$$
Roberto Bramati, Маттео Далла Ріва, Brian B. Luczak · Journal of Geometric Analysis · 2026
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.