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Large scale Sobolev inequalities on metric measure spaces and applications

Romain Tessera
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Abstract

We introduce a notion of "gradient at a given scale" of functions defined on a metric measure space. We then use it to define Sobolev inequalities at large scale and we prove their invariance under large-scale equivalence (maps that generalize the quasi-isometries). We prove that for a Riemmanian manifold satisfying a local Poincare 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 p \in [1,\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. 2007-02-25. Large scale Sobolev inequalities on metric measure spaces and applications. https://arxiv.org/abs/math/0702751

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