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

Haïm Brezis; Hoai-Minh Nguyen

doi:10.1016/j.crma.2010.11.026

Partial Differential Equations

On a new class of functions related to VMO

Presented by: Haïm Brezis

Haïm Brezis ¹; Hoai-Minh Nguyen ²

¹Rutgers University, Dept. of Math., Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
²Courant Institute, New York University, 251 Mercer St., New York, NY 10012, USA

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

Abstract (Original language)
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}

.

Received: 2010-11-23
Published online: 2011-01-03

DOI: 10.1016/j.crma.2010.11.026

Author's affiliations:

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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     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},
     year = {2011},
     publisher = {Elsevier},
     volume = {349},
     number = {3-4},
     doi = {10.1016/j.crma.2010.11.026},
     language = {en},
}

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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

References
Cited by

[1] J. Bourgain; H.-M. Nguyen A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 75-80

[2] H. Brezis; L. Nirenberg Degree theory and BMO, Part I: Compact manifolds without boundaries, Selecta Math., Volume 1 (1995), pp. 197-263

[3] H. Brezis, H.-M. Nguyen, On the distributional Jacobian of maps from $S^{N}$ into $S^{N}$ in fractional Sobolev and Hölder spaces, Ann. of Math., available online: 31 December 2009.

[4] F. John; L. Nirenberg On functions of bounded mean oscillation, Comm. Pure Appl. Math., Volume 14 (1961), pp. 415-426

[5] H.-M. Nguyen Some new characterizations of Sobolev spaces, J. Funct. Anal., Volume 237 (2006), pp. 689-720

[6] H.-M. Nguyen Optimal constant in a new estimate for the degree, J. d'Analyse Math., Volume 101 (2007), pp. 367-395

[7] H.-M. Nguyen, Some inequalities related to Sobolev norms, Calc. Var. Partial Differential Equations, published online: 12 October 2010, . | DOI

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