On a new class of functions related to <i>VMO</i>

Brezis, Haïm; Nguyen, Hoai-Minh

doi:10.1016/j.crma.2010.11.026

Partial Differential Equations

On a new class of functions related to VMO

Présenté par : Haïm Brezis

Brezis, Haïm ¹ ; Nguyen, Hoai-Minh 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

Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 157-160

Abstract (VO)
Résumé

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

I_{p} : = {g \in L^{1} (Ω; R); \underset{| g (x) - g (y) | > δ}{\int_{Ω} \int_{Ω}} \frac{1}{{| x - y |}^{d + p}} d x d y < + \infty \forall δ > 0}

where Ω is a bounded open subset of

R^{d}

,

d ⩾ 1

, and

p ⩾ 0

. In particular, we prove that

I_{d} \subset VMO

. This sharpens the well-known result stating that

W^{s, p} \subset VMO

for

0 < s < 1

and

s p = d

. Moreover, we establish that VMO is much bigger than

I_{d}

by showing that

VMO ⊄ I_{1}

. We also present some results when the double integral above is taken on the set

{(x, y) \in Ω \times Ω; | e^{i g (x)} - e^{i g (y)} | > δ}

.

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

I_{p} : = {g \in L^{1} (Ω; R); \underset{| g (x) - g (y) | > δ}{\int_{Ω} \int_{Ω}} \frac{1}{{| x - y |}^{d + p}} d x d y < + \infty \forall δ > 0},

où Ω est un ouvert borné de

R^{d}

,

d ⩾ 1

, et

p ⩾ 0

. En particulier, nous prouvons que

I_{d} \subset VMO

. Ceci améliore le résultat bien connu affirmant que

W^{s, p} \subset VMO

pour

0 < s < 1

et

s p = d

. D'autre part, nous prouvons que VMO est plus grand que

I_{d}

; en fait

VMO ⊄ I_{1}

. Nous présentons aussi des résultats lorsque l'intégrale double ci-dessus est prise sur l'ensemble

{(x, y) \in Ω \times Ω; | e^{i g (x)} - e^{i g (y)} | > δ}

.

Reçu le : 2010-11-23
Publié le : 2011-01-03

DOI : 10.1016/j.crma.2010.11.026

Affiliations des auteurs :

Brezis, Haïm ¹ ; Nguyen, Hoai-Minh 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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     author = {Brezis, Ha{\"\i}m and Nguyen, Hoai-Minh},
     title = {On a new class of functions related to {\protect\emph{VMO}}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {157--160},
     year = {2011},
     publisher = {Elsevier},
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Brezis, Haïm; Nguyen, Hoai-Minh. On a new class of functions related to VMO. Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 157-160. doi: 10.1016/j.crma.2010.11.026

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