An elementary proof of an inequality of Maz'ya involving $L^1$-vector fields
Résumé
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).
Domaines
Analyse classique [math.CA]
Origine : Fichiers produits par l'(les) auteur(s)
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