Comptes Rendus
Research article - Harmonic analysis
Quantitative suboptimal Sobolev embeddings
Antoine Detaille1, 2; Petru Mironescu1
1Universite Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208, 69622 Villeurbanne, France
2ETH Zürich, Department of Mathematics, Rämistrasse 101, Zürich 8092, Switzerland
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1363-1376

This article is part of the thematic issue From Generation to Generation: The Mathematical Legacy of Haïm Brezis coordinated by: Henri Berestycki et al..  

We present two new families of integral inequalities involving Sobolev seminorms associated with compact Sobolev embeddings. These inequalities quantify the fact that, on “many” small balls of a given domain, quantitative Sobolev embeddings are “much better” than predicted by scaling arguments.

On présente deux nouvelles familles d’inégalités intégrales impliquant des semi-normes de Sobolev, associées aux injections de Sobolev compactes. Ces inégalités quantifient le fait que, sur « un grand nombre » de boules contenues dans un domaine fixé, les inégalités de Sobolev quantitatives se comportent « bien mieux » que suggéré par un argument de mise à l’échelle.

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DOI: 10.5802/crmath.798
Classification: 46E35
Keywords: Sobolev embeddings, weak $L^p$-estimates
Mots-clés : Injections de Sobolev, estimées $L^p$ faibles

Antoine Detaille  1 , 2 ; Petru Mironescu  1

1 Universite Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208, 69622 Villeurbanne, France
2 ETH Zürich, Department of Mathematics, Rämistrasse 101, Zürich 8092, Switzerland
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Antoine Detaille; Petru Mironescu. Quantitative suboptimal Sobolev embeddings. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1363-1376. doi: 10.5802/crmath.798
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