Refined Sobolev inequalities on manifolds with ends

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 two-fold: on one hand, these inequalities are stable by multiplication by rapidly oscillating functions, much as the original ones, and on the other hand our Besov space is stable by spectral localization associated to the Laplace-Beltrami operator (while $L^p$ spaces, with $p \neq 2$, 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 self-adjoint operator on a measure space (Proposition 1).

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Accepted 24 February 2014

Published 27 October 2014