Large-scale Sobolev inequalities on metric measure spaces and applications

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 ≤ p ≤ ∞ is invariant under quasiisometry 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.

Isometric Group Actions on Hilbert Spaces: Growth of Cocycles

“…A locally compact, compactly generated group is either non-amenable or non-unimodular if and only if it is quasi-isometric to a graph with positive Cheeger constant (see [T2,Th. 2]).…”

The large-scale geometry of locally compact solvable groups

“…Regarding groups, this translates into focussing either on unimodular connected Lie groups, equipped with some leftinvariant Riemannian metric, or on finitely generated group equipped with some word metric. A unifying approach for metric measured spaces was given in [87,88], which allows in particular to treat all unimodular locally compact, compactly generated groups on the same footing.…”

“…It can be proved without too much difficulty that Sobolev inequalities such as (3.3) do not depend on the choice of µ and that they are stable under quasiisometry [19,87]. However, in complete generality, it is not known whether the asymptotic behavior of φ can be encoded in such functional inequalities.…”

Contracting automorphisms and Lp-cohomology in degree one

“…if and only if [T1,Proposition 11.9] G is either non-unimodular or non-amenable. This can be reformulated as -if G is non-compact, amenable and unimodular, then the topological vector space H 1 p (G) is non-Hausdorff (and in particular is non-zero); -otherwise, H 1 p (G)=H 1 p (G).…”