Comptes Rendus
no. 4
Partial differential equations/Functional analysis
Lp-Taylor approximations characterize the Sobolev space W1,p
[Les développements de Taylor-Lp caractérisent l'espace de Sobolev W1,p]

Présenté par : Haïm Brézis

Daniel Spector1, 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.

In this note, we introduce a variant of Calderón and Zygmund's notion of Lp-differentiability – an Lp-Taylor approximation. Our first result is that functions in the Sobolev space W1,p(RN) possess a first-order Lp-Taylor approximation. This is in analogy with Calderón and Zygmund's result concerning the Lp-differentiability of Sobolev functions. In fact, the main result we announce here is that the first-order Lp-Taylor approximation characterizes the Sobolev space W1,p(RN), and therefore implies Lp-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 Lp – un développement de Taylor-Lp. Notre premier résultat est que les fonctions de l'espace de Sobolev W1,p(RN) possèdent un développement de Taylor-Lp au premier ordre. C'est un analogue du résultat de Calderón et Zygmund concernant la différentiabilité Lp des fonctions de Sobolev. En fait, le résultat principal que nous annonçons ici est que le développement de Taylor-Lp au premier ordre caractérise l'espace de Sobolev W1,p(RN), et donc implique la différentibilité Lp. 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.01.010

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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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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  • Shibo Wang; Yuming Xing; Shengzhu Shi; Zhichang Guo A Taylor expansion-based texture and edge-preserving interpolation approach for arbitrary-scale image super-resolution, Pattern Recognition, Volume 169 (2026), p. 111965 | DOI:10.1016/j.patcog.2025.111965
  • Haim Brezis; Petru Mironescu Non-local approximations of the gradient, Confluentes Mathematici, Volume 15 (2024), pp. 27-44 | DOI:10.5802/cml.91 | Zbl:1550.46024
  • Daniel Spector New directions in harmonic analysis on L1, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 192 (2020), p. 20 (Id/No 111685) | DOI:10.1016/j.na.2019.111685 | Zbl:1437.42037
  • Augusto C. Ponce; Daniel Spector On formulae decoupling the total variation of BV functions, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 154 (2017), pp. 241-257 | DOI:10.1016/j.na.2016.08.028 | Zbl:1359.26011
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