On equivalence of super log Sobolev and Nash type inequalities
Volume 137 / 2014
Colloquium Mathematicum 137 (2014), 189-208 MSC: Primary 39B62. DOI: 10.4064/cm137-2-4
We prove the equivalence of Nash type and super log Sobolev 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 between super log Sobolev or Nash type inequalities and ultracontractivity. We discuss Davies–Simon's counterexample as the borderline case of this equivalence and related open problems.