Revista Matemática Iberoamericana

Large scale Sobolev inequalities on metric measure spaces and applications

Romain Tessera

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Abstract

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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 24, Number 3 (2008), 825-864.

Dates
First available in Project Euclid: 9 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1228834295

Mathematical Reviews number (MathSciNet)
MR2490163

Zentralblatt MATH identifier
1194.53036

Subjects
Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 22A10: Analysis on general topological groups

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

Citation

Tessera, Romain. Large scale Sobolev inequalities on metric measure spaces and applications. Rev. Mat. Iberoamericana 24 (2008), no. 3, 825--864. https://projecteuclid.org/euclid.rmi/1228834295


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