Weighted Nash Inequalities

Bolley, François; Gentil, Ivan; Bakry, Dominique

Bolley, François; Gentil, Ivan; Bakry, Dominique (2012), Weighted Nash Inequalities, Revista Matematica Iberoamericana, 28, 3, p. 879-906. 10.4171/RMI/695

Type

Article accepté pour publication ou publié

External document link

https://arxiv.org/abs/1004.3456v2

Date

2012

Journal name

Revista Matematica Iberoamericana

Volume

Number

Publisher

Consejo Superior de Investigaciones Científicas

Pages

879-906

Abstract (EN)

Nash or Sobolev inequalities are known to be equivalent to ultracontractive properties of Markov semigroups, hence to uniform bounds on their kernel densities. In this work we present a simple and extremely general method, based on weighted Nash inequalities, to obtain non-uniform bounds on the kernel densities. Such bounds imply a control on the trace or the Hilbert-Schmidt norm of the heat kernels. We illustrate the method on the heat kernel on $\dR$ naturally associated with the measure with density $C_a\exp(-|x|^a)$, with $1

Subjects / Keywords

Super-Poincaré inequality; Ultracontractivity; Heat kernel; Nash inequality