Weighted Nash Inequalities

Dominique Bakry (1, 2) , François Bolley (3) , Ivan Gentil (3)

1 IUF - Institut universitaire de France
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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

Domaines: Probabilités [math.PR]

Dominique Bakry, François Bolley, Ivan Gentil. Weighted Nash Inequalities. 2010. ⟨hal-00474543v1⟩