A surprising formula for Sobolev norms

Haïm Brezis; Jean van Schaftingen; Po Lam Yung

doi:10.1073/pnas.2025254118

A surprising formula for Sobolev norms

Haïm Brezis^*, Jean van Schaftingen, Po Lam Yung

^*Corresponding author for this work

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

35 Citations (Scopus)

Abstract

We establish the equivalence between the Sobolev seminorm k∇uk_Lp and a quantity obtained when replacing strong L^p by weak Lp in the Gagliardo seminorm |u|_Ws,p computed at s = 1. As corollaries we derive alternative estimates in some exceptional cases (involving W^1,1) where the “anticipated” fractional Sobolev and Gagliardo–Nirenberg inequalities fail.

Original language	English
Article number	e2025254118
Journal	Proceedings of the National Academy of Sciences of the United States of America
Volume	118
Issue number	8
DOIs	https://doi.org/10.1073/pnas.2025254118
Publication status	Published - 23 Feb 2021

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abstract = "We establish the equivalence between the Sobolev seminorm k∇ukLp and a quantity obtained when replacing strong Lp by weak Lp in the Gagliardo seminorm |u|Ws,p computed at s = 1. As corollaries we derive alternative estimates in some exceptional cases (involving W1,1) where the “anticipated” fractional Sobolev and Gagliardo–Nirenberg inequalities fail.",

keywords = "Fractional Gagliardo, Fractional Sobolev space, Marcinkiewicz space, Nirenberg interpolation inequalities",

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AU - Brezis, Haïm

AU - van Schaftingen, Jean

AU - Yung, Po Lam

PY - 2021/2/23

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N2 - We establish the equivalence between the Sobolev seminorm k∇ukLp and a quantity obtained when replacing strong Lp by weak Lp in the Gagliardo seminorm |u|Ws,p computed at s = 1. As corollaries we derive alternative estimates in some exceptional cases (involving W1,1) where the “anticipated” fractional Sobolev and Gagliardo–Nirenberg inequalities fail.

AB - We establish the equivalence between the Sobolev seminorm k∇ukLp and a quantity obtained when replacing strong Lp by weak Lp in the Gagliardo seminorm |u|Ws,p computed at s = 1. As corollaries we derive alternative estimates in some exceptional cases (involving W1,1) where the “anticipated” fractional Sobolev and Gagliardo–Nirenberg inequalities fail.

KW - Fractional Gagliardo

KW - Fractional Sobolev space

KW - Marcinkiewicz space

KW - Nirenberg interpolation inequalities

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U2 - 10.1073/pnas.2025254118

DO - 10.1073/pnas.2025254118

M3 - Article

SN - 0027-8424

VL - 118

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

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M1 - e2025254118

ER -

A surprising formula for Sobolev norms

Abstract

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