ON EQUIVALENCE OF SUPER LOG SOBOLEV AND NASH TYPE INEQUALITIES (old title: Super Logarithmic Sobolev inequalities and Nash-type inequalities for sub-markovian symmetric semigroups)

Marco Biroli (1) , Patrick Maheux (2)

1 DIPARTIMENTO DI MATEMATICA - Dipartimento di Matematica
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

We prove the equivalence of Nash type and super logSobolev inequalities for Dirichlet forms. We also show that both inequalities are equivalent to Orlicz-Sobolev type inequalities. No ultracontractivity of the semigroup is assumed. It is known that there is no equivalence be- tween super log Sobolev or Nash type inequalities and ultracontractiv- ity. We discuss Davies-Simon’s counterexample as borderline case of this equivalence and related open problems.

Mots clés: Logarithmic Sobolev inequalities; Nash-type inequality; sub-markovian semigroup; Orlicz-Sobolev inequality

Domaines: Analyse classique [math.CA]

Marco Biroli, Patrick Maheux. ON EQUIVALENCE OF SUPER LOG SOBOLEV AND NASH TYPE INEQUALITIES (old title: Super Logarithmic Sobolev inequalities and Nash-type inequalities for sub-markovian symmetric semigroups). 2010. ⟨hal-00465177v2⟩