A surprising formula for Sobolev norms

被引：39

作者：

Brezis, Haim ^{[1
,2
,3
,4
]}

Van Schaftingen, Jean ^{[5
]}

Yung, Po-Lam ^{[6
,7
]}

机构：

[1] Rutgers State Univ, Dept Math, Piscataway, NJ 08854 USA

[2] Technion Israel Inst Technol, Dept Math, IL-32000 Haifa, Israel

[3] Technion Israel Inst Technol, Dept Comp Sci, IL-32000 Haifa, Israel

[4] Sorbonne Univ, Lab Jacques Louis Lions, F-75005 Paris, France

[5] Catholic Univ Louvain, Inst Rech Math & Phys, B-1348 Louvain La Neuve, Belgium

[6] Australian Natl Univ, Math Sci Inst, Canberra, ACT 2601, Australia

[7] Chinese Univ Hong Kong, Dept Math, Ma Liu Shui, Hong Kong, Peoples R China

来源：

PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA | 2021年 / 118卷 / 08期

关键词：

fractional Sobolev space; Marcinkiewicz space; fractional Gagliardo-Nirenberg interpolation inequalities; INEQUALITIES;

D O I：

10.1073/pnas.2025254118

中图分类号：

O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];

学科分类号：

07 ; 0710 ; 09 ;

摘要：

We establish the equivalence between the Sobolev seminorm parallel to del u parallel to(Lp) and a quantity obtained when replacing strong Lp by weak L-p in the Gagliardo seminorm vertical bar u vertical bar W-s,W-p computed at s = 1. As corollaries we derive alternative estimates in some exceptional cases (involving W-1,W-1) where the "anticipated" fractional Sobolev and Gagliardo-Nirenberg inequalities fail.

引用

页数：6

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