An elementary proof of an inequality of Maz'ya involving $L^1$-vector fields

Pierre Bousquet (1) , Petru Mironescu (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 EDPA - Équations aux dérivées partielles, analyse

We give a short elementary proof of the inequality \begin{equation*} \| D (-\Delta)^{-1} {\mathbf f}\|_{L^q(|x|^{n(q-1)-q}\, dx)}\le c(\|{\mathbf f}\|_{L^1}+\|\nabla (-\Delta)^{-1}\,\text{div}\, {\mathbf f}\|_{L^1}), \end{equation*} $\forall\, {\mathbf f}\in L^1({\mathbb R}^n ; {\mathbb R}^n)$, $\forall\, 1\le q$<$n/(n-1)$, essentially established by Maz'ya (J. Eur. Math. Soc. 2010).

Mots clés: Analysis of vector fields

Domaines: Analyse classique [math.CA]

Pierre Bousquet, Petru Mironescu. An elementary proof of an inequality of Maz'ya involving $L^1$-vector fields. Contemporary mathematics, 2011, 540 (Nonlinear partial differential equations), pp.59-63. ⟨10.1090/conm/540/10659⟩. ⟨hal-00425043⟩