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Large scale Sobolev inequalities on metric measure spaces and applications. (English) Zbl 1194.53036

Summary: 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.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
22D05 General properties and structure of locally compact groups
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
60G50 Sums of independent random variables; random walks
20F65 Geometric group theory
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