<p>We show that there are harmonic functions on a ball <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\mathbb {B}_n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, that are continuous, and even Hölder continuous, up to the boundary but not in the Sobolev space <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$H^s(\mathbb {B}_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">B</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <i>s</i> 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 <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> we exploit certain series of spherical harmonics.</p>

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

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2026年5月11日

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Roberto Bramati,
Matteo Dalla Riva,
Brian Luczak

摘要

We show that there are harmonic functions on a ball ${\mathbb {B}_n}$ $B_{n}$ of $\mathbb {R}^n$ $R^{n}$ , $n\ge 2$ $n \geq 2$ , 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 $n\ge 2$ $n \geq 2$ we exploit certain series of spherical harmonics.

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

摘要