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On equivalence of super log Sobolev and Nash type inequalities

Volume 137 / 2014

Marco Biroli, Patrick Maheux 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.


  • Marco BiroliDipartimento di Matematica “F. Brioschi”
    Politecnico di Milano
    Piazza Leonardo da Vinci 32
    20133, Milano, Italy
  • Patrick MaheuxFédération Denis Poisson
    (MAPMO, UMR-CNRS 7349)
    Département de Mathématiques
    Université d'Orléans
    45067 Orléans Cedex 2, France

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