From U-bounds to isoperimetry with applications to H-type groups

Abstract

In this paper we study U -bounds in relation to L 1 -type coercive inequalities and isoperimetric problems for a class of probability measures on a general metric space (R N , d). We prove the equivalence of an isoperimetric inequality with several other coercive inequalities in this general framework. The usefulness of our approach is illustrated by an application to the setting of H-type groups, and an extension to infinite dimensions.

Markov semigroups with hypocoercive-type generator in infinite dimensions: Ergodicity and smoothing

“…with a convention x >> y meaning x ≥ Cy 2 + C ′ with some constants C, C ′ ∈ [1, ∞) sufficiently large and possibly dependent on n, but not on k. In this way we get (15).…”

Coercive inequalities in higher-dimensional anisotropic heisenberg group

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“…We point out, that in [21], M. Ledoux made a connection between the Logarithmic Sobolev inequality and the isoperimetric problem. (See also: [1,3,18])…”