Quantitative suboptimal Sobolev embeddings

Antoine Detaille; Petru Mironescu

doi:10.5802/crmath.798

Research article - Harmonic analysis

Quantitative suboptimal Sobolev embeddings

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

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 edited by Jean-Michel Coron et al..

Abstract (Original language)
Résumé

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.

Received: 2025-06-12
Revised: 2025-09-18
Accepted: 2025-09-26
Published online: 2025-11-03

DOI: 10.5802/crmath.798

Classification: 46E35
Keywords: Sobolev embeddings, weak $L^p$-estimates
Mots-clés : Injections de Sobolev, estimées $L^p$ faibles

Author's affiliations:

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

@article{CRMATH_2025__363_G12_1363_0,
     author = {Antoine Detaille and Petru Mironescu},
     title = {Quantitative suboptimal {Sobolev} embeddings},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1363--1376},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.798},
     language = {en},
}

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%A Petru Mironescu
%T Quantitative suboptimal Sobolev embeddings
%J Comptes Rendus. Mathématique
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Antoine Detaille; Petru Mironescu. Quantitative suboptimal Sobolev embeddings. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1363-1376. doi: 10.5802/crmath.798

References
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