Partial Differential Equations
On a new class of functions related to VMO
Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 157-160.

In this Note, we compare the space VMO and the spaces

 $Ip:={g∈L1(Ω;R);∫Ω∫Ω|g(x)−g(y)|>δ1|x−y|d+pdxdy<+∞∀δ>0}$
where Ω is a bounded open subset of $Rd$, $d⩾1$, and $p⩾0$. In particular, we prove that $Id⊂VMO$. This sharpens the well-known result stating that $Ws,p⊂VMO$ for $0 and $sp=d$. Moreover, we establish that VMO is much bigger than $Id$ by showing that $VMO⊄I1$. We also present some results when the double integral above is taken on the set ${(x,y)∈Ω×Ω;|eig(x)−eig(y)|>δ}$.

Dans cette Note, nous comparons l'espace VMO et les espaces

 $Ip:={g∈L1(Ω;R);∫Ω∫Ω|g(x)−g(y)|>δ1|x−y|d+pdxdy<+∞∀δ>0},$
Ω est un ouvert borné de $Rd$, $d⩾1$, et $p⩾0$. En particulier, nous prouvons que $Id⊂VMO$. Ceci améliore le résultat bien connu affirmant que $Ws,p⊂VMO$ pour $0 et $sp=d$. D'autre part, nous prouvons que VMO est plus grand que $Id$ ; en fait $VMO⊄I1$. Nous présentons aussi des résultats lorsque l'intégrale double ci-dessus est prise sur l'ensemble ${(x,y)∈Ω×Ω;|eig(x)−eig(y)|>δ}$.

Published online:
DOI: 10.1016/j.crma.2010.11.026

Haïm Brezis 1; Hoai-Minh Nguyen 2

1 Rutgers University, Dept. of Math., Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
2 Courant Institute, New York University, 251 Mercer St., New York, NY 10012, USA
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Haïm Brezis; Hoai-Minh Nguyen. On a new class of functions related to VMO. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 157-160. doi : 10.1016/j.crma.2010.11.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.026/

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