Comptes Rendus
Partial differential equations/Functional analysis
Lp-Taylor approximations characterize the Sobolev space W1,p
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.

Published online:
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
     author = {Daniel Spector},
     title = {$ {L}^{p}${-Taylor} approximations characterize the {Sobolev} space $ {W}^{1,p}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {327--332},
     publisher = {Elsevier},
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     number = {4},
     year = {2015},
     doi = {10.1016/j.crma.2015.01.010},
     language = {en},
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JO  - Comptes Rendus. Mathématique
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PB  - Elsevier
DO  - 10.1016/j.crma.2015.01.010
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ID  - CRMATH_2015__353_4_327_0
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%R 10.1016/j.crma.2015.01.010
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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.

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