$ {L}^{p}$-Taylor approximations characterize the Sobolev space $ {W}^{1,p}$

Daniel Spector

doi:10.1016/j.crma.2015.01.010

Partial differential equations/Functional analysis

L^{p}

-Taylor approximations characterize the Sobolev space

W^{1, p}

Presented by: Haïm Brézis

Daniel Spector ^1,²

¹ Technion – Israel Institute of Technology, Department of Mathematics, Haifa, Israel
² National Chiao Tung University, Department of Applied Mathematics, Hsinchu, Taiwan

Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 327-332.

Abstract (Original language)
Résumé

In this note, we introduce a variant of Calderón and Zygmund's notion of $L^{p}$ -differentiability – an $L^{p}$ -Taylor approximation. Our first result is that functions in the Sobolev space $W^{1, p} (R^{N})$ possess a first-order $L^{p}$ -Taylor approximation. This is in analogy with Calderón and Zygmund's result concerning the $L^{p}$ -differentiability of Sobolev functions. In fact, the main result we announce here is that the first-order $L^{p}$ -Taylor approximation characterizes the Sobolev space $W^{1, p} (R^{N})$ , and therefore implies $L^{p}$ -differentiability. Our approach establishes connections between some characterizations of Sobolev spaces due to Swanson using Calderón–Zygmund classes with others due to Bourgain, Brézis, and Mironescu using nonlocal functionals with still others of the author and Mengesha using nonlocal gradients. That any two characterizations of Sobolev spaces are related is not surprising; however, one consequence of our analysis is a simple condition for determining whether a function of bounded variation is in a Sobolev space.

Dans cette note, nous introduisons une variante de la notion de Calderón et Zygmund de différentiabilité $L^{p}$ – un développement de Taylor- $L^{p}$ . Notre premier résultat est que les fonctions de l'espace de Sobolev $W^{1, p} (R^{N})$ possèdent un développement de Taylor- $L^{p}$ au premier ordre. C'est un analogue du résultat de Calderón et Zygmund concernant la différentiabilité $L^{p}$ des fonctions de Sobolev. En fait, le résultat principal que nous annonçons ici est que le développement de Taylor- $L^{p}$ au premier ordre caractérise l'espace de Sobolev $W^{1, p} (R^{N})$ , et donc implique la différentibilité $L^{p}$ . Notre approche établit des liens entre les caractérisations des espaces de Sobolev dues à Swanson, qui utilisent les classes de Calderón–Zygmund, celles dues à Bourgain, Brézis et Mironescu, qui utilisent des fonctionnelles non locales, et celles dues à l'auteur et à Mengesha, qui utilisent des gradients non locaux. Que les différentes caractérisations des espaces de Sobolev soient reliées n'est pas surprenant ; cependant, notre analyse donne une condition simple pour déterminer si une fonction à variation bornée est dans un espace de Sobolev.

Received: 2014-06-08
Accepted: 2015-01-26
Published online: 2015-02-13

DOI: 10.1016/j.crma.2015.01.010

Author's affiliations:

Daniel Spector ^{1, 2}

¹ Technion – Israel Institute of Technology, Department of Mathematics, Haifa, Israel
² National Chiao Tung University, Department of Applied Mathematics, Hsinchu, Taiwan

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     title = {$ {L}^{p}${-Taylor} approximations characterize the {Sobolev} space $ {W}^{1,p}$},
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Daniel Spector. $ {L}^{p}$-Taylor approximations characterize the Sobolev space $ {W}^{1,p}$. Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 327-332. doi : 10.1016/j.crma.2015.01.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.01.010/

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