On some inequalities of Bourgain, Brezis, Maz'ya, and Shaposhnikova related to $L^1$ vector fields - Archive ouverte HAL Access content directly
Journal Articles Comptes rendus de l'Académie des sciences. Série I, Mathématique Year : 2010

On some inequalities of Bourgain, Brezis, Maz'ya, and Shaposhnikova related to $L^1$ vector fields

Abstract

Bourgain and Brezis (J. Amer. Math. Soc. 2003) established, for maps $f\in L^n({\mathbb T}^n)$ with zero average, the existence of a solution $\vec{Y}\in W^{1,n}\cap L^\infty$ of (1) div $\vec{Y}=f$. Maz'ya (Contemp. Math. vol. 445) proved that if, in addition $f\in H^{n/2-1}({\mathbb T}^n)$, then (1) can be solved in $H^{n/2}\cap L^\infty$. Their arguments are quite different. We present an elementary property of the biharmonic operator in two dimensions. This property unifies, in two dimensions, the two approaches, and implies another (apparently unrelated) estimate of Maz'ya and Shaposhnikova (Sobolev spaces in mathematics I, 2009). We discuss higher dimensional analogs of the above results.
Fichier principal
Vignette du fichier
cras_l1_fields_20100323.pdf (331.54 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-00747678 , version 1 (31-10-2012)

Identifiers

Cite

Petru Mironescu. On some inequalities of Bourgain, Brezis, Maz'ya, and Shaposhnikova related to $L^1$ vector fields. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2010, 348 (9-10), pp.513-515. ⟨10.1016/j.crma.2010.03.019⟩. ⟨hal-00747678⟩
258 View
177 Download

Altmetric

Share

Gmail Facebook X LinkedIn More