Large scale Sobolev inequalities on metric measure spaces and applications

Details Large scale Sobolev inequalities on metric measure spaces and applications Large scale Sobolev inequalities on metric measure spaces and applications

2007-02-25

arxiv.org/abs/math/0702751v1

Mathematics

Metric Geometry

43 pages

Scientific paper

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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