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Sobolev inequalities involving divergence free maps. (English) Zbl 0778.42011
Le but de ce papier est de donner une démonstration élémentaire de l’inégalité ci-dessous. On considère \(f,h \in H^ 1(\mathbb{R}^ n)\) et \(g \in L^ 2(\mathbb{R}^ n;\mathbb{R}^ n)\) qui vérifie \(\nabla \cdot g=0\). On suppose aussi que \[ A^ 2:=\sup_{z,r}r^{2-n} \int_{B(z,r)}| \nabla h |^ 2dx<\infty, \] où \(B(z,r)\) est la boule de \(\mathbb{R}^ n\), centrée en \(z\), de rayon \(r\). On prouve alors qu’il existe une constante \(c=c(n)\), indépendante de \(f,g\) et \(h\), telle que \[ \left| \int_{\mathbb{R}^ n}fg \cdot \nabla h dx \right| \leq cA \| \nabla f \|_{L^ 2(\mathbb{R}^ n)}\| g \|_{L^ 2(\mathbb{R}^ n)}. \] On obtient aussi une version “avec poids” de cette inégalité.

MSC:
42B25 Maximal functions, Littlewood-Paley theory
26D10 Inequalities involving derivatives and differential and integral operators
42B30 \(H^p\)-spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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