Mathematical Research Letters

Volume 21 (2014)

Number 4

Refined Sobolev inequalities on manifolds with ends

Pages: 633 – 675

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n4.a3

Authors

Jean-Marc Bouclet (Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France)

Yannick Sire (Centre de Mathématiques et Informatique (CMI), Institut de Mathématiques, Université Aix-Marseille, France)

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 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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