Sobolev spaces revisited

Haïm Brezis; Andreas Seeger; Jean van Schaftingen; Po Lam Yung

doi:10.4171/RLM/976

Sobolev spaces revisited

Haïm Brezis, Andreas Seeger, Jean van Schaftingen, Po Lam Yung

Research output: Contribution to journal › Article › peer-review

25 Citations (Scopus)

Abstract

We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on Rⁿ, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis, and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies, and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo–Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the L^p norm of functions in L^p(Rⁿ).

Original language	English
Pages (from-to)	413-437
Number of pages	25
Journal	Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni
Volume	33
Issue number	2
DOIs	https://doi.org/10.4171/RLM/976
Publication status	Published - 31 Aug 2022

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T1 - Sobolev spaces revisited

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AU - van Schaftingen, Jean

AU - Yung, Po Lam

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KW - BBM formula

KW - Gagliardo seminorms

KW - Gagliardo–Nirenberg interpolation

KW - Sobolev spaces

KW - bounded variation

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Sobolev spaces revisited

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