# Revista Matemática Iberoamericana

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**Volume 24, Issue 3, 2008, pp. 825–864**

**DOI: 10.4171/RMI/557**

Published online: 2008-12-31

Large-scale Sobolev inequalities on metric measure spaces and applications

Romain Tessera^{[1]}(1) Université Paris-Sud, Orsay, France

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.

*Keywords: *Large-scale analysis on metric spaces, coarse equivalence, symmetric random walks on groups, Sobolev inequalities, isoperimetry

Tessera Romain: Large-scale Sobolev inequalities on metric measure spaces and applications. *Rev. Mat. Iberoam.* 24 (2008), 825-864. doi: 10.4171/RMI/557