<p>In this paper we consider the <i>N</i>-dimensional Euclidean Onofri inequality proved by del Pino and Dolbeault (Int Math Res Not IMRN 15:3600–3611, 2013) for smooth compactly supported functions in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2935_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <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="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2935_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <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 extend the inequality to a suitable weighted Sobolev space, although no clear connection with standard Sobolev spaces on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2935_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {S}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> through stereographic projection is present, except for the planar case. Moreover, in any dimension <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2935_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <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 show that the Euclidean Onofri inequality is equivalent to the logarithmic Moser–Trudinger inequality with sharp constant proved by Carleson and Chang (Bull Sci Math 110(2):113–127, 1986) for balls in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2935_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <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>.</p>

On the equivalence between an Onofri-type inequality by Del Pino–Dolbeault and the sharp logarithmic Moser–Trudinger inequality

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2025年2月6日

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Natalino Borgia,
Silvia Cingolani,
Gabriele Mancini

摘要

In this paper we consider the N-dimensional Euclidean Onofri inequality proved by del Pino and Dolbeault (Int Math Res Not IMRN 15:3600–3611, 2013) for smooth compactly supported functions in $\mathbb {R}^N$ $R^{N}$ , $N \ge 2$ $N \geq 2$ . We extend the inequality to a suitable weighted Sobolev space, although no clear connection with standard Sobolev spaces on $\mathbb {S}^N$ $S^{N}$ through stereographic projection is present, except for the planar case. Moreover, in any dimension $N \ge 2$ $N \geq 2$ , we show that the Euclidean Onofri inequality is equivalent to the logarithmic Moser–Trudinger inequality with sharp constant proved by Carleson and Chang (Bull Sci Math 110(2):113–127, 1986) for balls in $\mathbb {R}^N$ $R^{N}$ .