Comptes Rendus
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

where Ω is a bounded open subset of Rd, d1, and p0. In particular, we prove that IdVMO. This sharpens the well-known result stating that Ws,pVMO for 0<s<1 and sp=d. Moreover, we establish that VMO is much bigger than Id by showing that VMOI1. 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

Ω est un ouvert borné de Rd, d1, et p0. En particulier, nous prouvons que IdVMO. Ceci améliore le résultat bien connu affirmant que Ws,pVMO pour 0<s<1 et sp=d. D'autre part, nous prouvons que VMO est plus grand que Id ; en fait VMOI1. 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
     author = {Ha{\"\i}m Brezis and Hoai-Minh Nguyen},
     title = {On a new class of functions related to {\protect\emph{VMO}}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {157--160},
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%A Hoai-Minh Nguyen
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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.

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