<p>In this paper we prove sharp multipolar Hardy-type inequalities in the Riemannian <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2206_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p-$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>setting for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2206_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> using the method of super-solutions and fundamental results from comparison theory on manifolds, thus generalizing the work in [<CitationRef CitationID="CR14">14</CitationRef>, <CitationRef CitationID="CR16">16</CitationRef>] for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2206_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We emphasize that when we restrict to Cartan-Hadamard manifolds, the inequalities improve in the case <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2206_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$2<p<N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> compared to the case <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2206_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> since we obtain positive remainder terms which are controlled by curvature estimates. In the end, we treat the cases of positive and negative constant sectional curvature.</p>

Sharp multipolar $L^p$-Hardy-type inequalities on Riemannian manifolds

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2025年9月27日

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Cristian Ciulică,
Teodor Rugină

摘要

In this paper we prove sharp multipolar Hardy-type inequalities in the Riemannian $L^p-$ $L^{p} -$ setting for $p\ge 2$ $p \geq 2$ using the method of super-solutions and fundamental results from comparison theory on manifolds, thus generalizing the work in [14, 16] for $p=2$ $p = 2$ . We emphasize that when we restrict to Cartan-Hadamard manifolds, the inequalities improve in the case $2<p<N$ $2 < p < N$ compared to the case $p=2$ $p = 2$ since we obtain positive remainder terms which are controlled by curvature estimates. In the end, we treat the cases of positive and negative constant sectional curvature.

Sharp multipolar \(L^p\)-Hardy-type inequalities on Riemannian manifolds

摘要