New Proofs of Davies-Simon's Theorems about Ultracontractivity and Logarithmic Sobolev Inequalities related to Nash Type Inequalities

Patrick Maheux (1)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

We present new proofs of two theorems of E.B. Davies and B. Simon about ultracontractivity property for of semigroups of operators and logarithmic Sobolev inequalities with parameter (LSIWP for short) satisfied by the generator of the semigroup. In our proof, we use neither the $L^p$ version of the $LSIWP$ nor Stein's interpolation. Our tool is Nash type inequality. We also present new results. In particular, we show that Nash type inequality and LSIWP are equivalent.

Mots clés: ultracontractivity property; logarithmic Sobolev inequality with parameter; Nash type inequality; semigroups of operators; Dirichlet form; Heat kernel; infinite Torus.; infinite Torus

Domaines: Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]

Patrick Maheux. New Proofs of Davies-Simon's Theorems about Ultracontractivity and Logarithmic Sobolev Inequalities related to Nash Type Inequalities. 2006. ⟨hal-00091044⟩