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arXiv · 1308.0853

Refined Sobolev inequalities on manifolds with ends

Jean-Marc Bouclet·Yannick Sire
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Abstract

By considering a suitable Besov type norm, we obtain refined Sobolev inequalities on a family of Riemannian manifolds with (possibly exponentially large) ends. The interest is twofold: on one hand, these inequalities are stable by multiplication by rapidly oscillating functions, much as the original ones \cite{GMO}, and on the other hand our Besov space is stable by spectral localization associated to the Laplace-Beltrami operator (while $ L^p $ spaces, with $ p \ne 2 $, are in general not preserved by such localizations on manifolds with exponentially large ends). We also prove an abstract version of refined Sobolev inequalities for any selfadjoint operator on a measure space (Proposition \ref{general}).

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Jean-Marc Bouclet, Yannick Sire. 2013-08-04. Refined Sobolev inequalities on manifolds with ends. https://arxiv.org/abs/1308.0853

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