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

Petru Mironescu (1)

1 EDPA - Équations aux dérivées partielles, analyse

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.

Mots clés: critical spaces; limiting Sobolev embeddings; divergence equation

Domaines: Analyse classique [math.CA]

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⟩