Revista Matemática Iberoamericana

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Volume 28, Issue 3, 2012, pp. 879–906
DOI: 10.4171/RMI/695

Published online: 2012-07-16

Weighted Nash inequalities

Dominique Bakry[1], François Bolley[2], Ivan Gentil[3] and Patrick Maheux[4]

(1) Université Paul Sabatier, Toulouse, France
(2) Université de Paris Dauphine, France
(3) Université Claude Bernard Lyon 1, Villeurbanne, France
(4) Université d'Orléans, France

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 paper, following work of F.-Y. Wang, we present a simple and extremely general method, based on weighted Nash inequalities, for obtaining non-uniform bounds on kernel densities. Such bounds imply control of the trace or the Hilbert–Schmidt norm of the heat kernels. We illustrate the method on the heat kernel on $\mathbb{R}$ naturally associated with the measure with density $C_a\exp(-|x|^a)$, with $1< a < 2 $, for which uniform bounds are known not to hold.

Keywords: Nash inequality, super-Poincaré inequality, heat kernel, ultracontractivity

Bakry Dominique, Bolley François, Gentil Ivan, Maheux Patrick: Weighted Nash inequalities. Rev. Mat. Iberoam. 28 (2012), 879-906. doi: 10.4171/RMI/695