Recovering functions via doubly homogeneous nonlocal gradients

Stefano Buccheri; Augusto C. Ponce

doi:10.5802/crmath.800

Research article - Partial differential equations

Recovering functions via doubly homogeneous nonlocal gradients

Stefano Buccheri ¹ ; Augusto C. Ponce ²

¹ Università degli Studi di Napoli Federico II, Dipartimento di Matematica e Applicazioni R. Caccioppoli, Via Vicinale dell’Infermeria 58, 80125 Napoli, Italy
² Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du cyclotron 2, L7.01.02, 1348 Louvain-la-Neuve, Belgium

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1377-1406

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 investigate a class of nonlocal gradients featuring distinct homogeneities at zero and infinity. We establish a representation formula for such doubly homogeneous operators and derive associated Sobolev-type inequalities. We also propose open questions linked to our results, suggesting directions for future research inspired by the work of Haïm Brezis.

Nous étudions une classe de gradients non locaux présentant des homogénéités distinctes en zéro et à l’infini. Nous établissons une formule de représentation pour ces opérateurs doublement homogènes et en déduisons des inégalités de Sobolev associées. Nous proposons également des questions ouvertes liées à nos résultats, suggérant des directions de recherche inspirées par les travaux de Haïm Brezis.

Received: 2025-06-15
Revised: 2025-09-22
Accepted: 2025-10-07
Published online: 2025-11-03

DOI: 10.5802/crmath.800

Classification: 26A33, 47G20, 42A38, 47A67
Keywords: Riesz fractional gradient, nonlocal gradient, representation formula, Sobolev inequality, Fourier transform, Brezis-type problems
Mots-clés : Gradient fractionnaire de Riesz, gradient non local, formule de représentation, inégalité de Sobolev, transformée de Fourier, problèmes de type Brezis

Author's affiliations:

Stefano Buccheri ¹; Augusto C. Ponce ²

¹ Università degli Studi di Napoli Federico II, Dipartimento di Matematica e Applicazioni R. Caccioppoli, Via Vicinale dell’Infermeria 58, 80125 Napoli, Italy
² Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du cyclotron 2, L7.01.02, 1348 Louvain-la-Neuve, Belgium

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Stefano Buccheri and Augusto C. Ponce},
     title = {Recovering functions via doubly homogeneous nonlocal gradients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1377--1406},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.800},
     language = {en},
}

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Stefano Buccheri; Augusto C. Ponce. Recovering functions via doubly homogeneous nonlocal gradients. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1377-1406. doi: 10.5802/crmath.800

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