<p>In this paper, a Riesz variational characterization of the first-order Sobolev space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1940_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$W_p^1({\mathbb {S}}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>W</mi> <mi>p</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1940_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$d<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> on the unit sphere <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1940_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {S}}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> is established.</p>

A Riesz Variational Characterization of Sobolev Spaces on the Sphere

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发表日期：

2025年2月21日

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Jiaxin Geng,
Yun Ling,
Jiansong Li,
Heping Wang

摘要

In this paper, a Riesz variational characterization of the first-order Sobolev space $W_p^1({\mathbb {S}}^d)$ $W_{p}^{1} (S^{d})$ , $d<p<\infty $ $d < p < \infty$ on the unit sphere ${\mathbb {S}}^d$ $S^{d}$ is established.