Open Access
November, 2008 Large scale Sobolev inequalities on metric measure spaces and applications
Romain Tessera
Rev. Mat. Iberoamericana 24(3): 825-864 (November, 2008).


For functions on a metric measure space, we introduce a notion of ``gradient at a given scale''. This allows us to define Sobolev inequalities at a given scale. We prove that satisfying a Sobolev inequality at a large enough scale is invariant under large-scale equivalence, a metric-measure version of coarse equivalence. We prove that for a Riemmanian manifold satisfying a local Poincaré inequality, our notion of Sobolev inequalities at large scale is equivalent to its classical version. These notions provide a natural and efficient point of view to study the relations between the large time on-diagonal behavior of random walks and the isoperimetry of the space. Specializing our main result to locally compact groups, we obtain that the $L^p$-isoperimetric profile, for every $1\leq p\leq \infty$ is invariant under quasi-isometry between amenable unimodular compactly generated locally compact groups. A qualitative application of this new approach is a very general characterization of the existence of a spectral gap on a quasi-transitive measure space $X$, providing a natural point of view to understand this phenomenon.


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Romain Tessera . "Large scale Sobolev inequalities on metric measure spaces and applications." Rev. Mat. Iberoamericana 24 (3) 825 - 864, November, 2008.


Published: November, 2008
First available in Project Euclid: 9 December 2008

zbMATH: 1194.53036
MathSciNet: MR2490163

Primary: 20F65 , 22A10

Keywords: coarse equivalence , Isoperimetry , large-scale analysis on metric spaces , Sobolev inequalities , symmetric random walks on groups

Rights: Copyright © 2008 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.24 • No. 3 • November, 2008
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