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.
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页数：6
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