Riesz transforms on connected sums

Carron, Gilles

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2329-2343.

Soit $M_0$ une variété riemannienne complète à courbure de Ricci bornée inférieurement et qui vérifie l’inégalité Sobolev de dimension $\nu >3$. Si $M$ est une variété riemannienne complète isométrique à $M_0$ en dehors d’un compact et si $p\in (\nu /(\nu -1),\nu )$ alors lorsque la transformée de Riesz est bornée sur $L^p\left(M_0\right)$ elle est également bornée sur $L^p\left(M\right)$.

Assume that $M_0$ is a complete Riemannian manifold with Ricci curvature bounded from below and that $M_0$ satisfies a Sobolev inequality of dimension $\nu >3$. Let $M$ be a complete Riemannian manifold isometric at infinity to $M_0$ and let $p\in (\nu /(\nu -1),\nu )$. The boundedness of the Riesz transform of $L^p\left(M_0\right)$ then implies the boundedness of the Riesz transform of $L^p\left(M\right)$

Classification : 58J37, 58J35, 42B20
Keywords: Riesz transform, Sobolev inequalities
Mot clés : transformée de Riesz, inégalités de Sobolev

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