Nash-type inequalities and decay of semigroups of operators

In that paper, we prove an equivalence between Nash-type inequalities and an exponential decay (in the sense of the definition 2.2) for symmetric submarkovian semigroups. This exponential decay generalizes the notion of spectral gap where this number is replaced by a function. We discuss different formulations of the decay associated to the usual Nash inequality in terms of Lyapunov-type functional. We apply this to different classes of ultracontractive semigroups as well as non-ultracontractive semigroups. In particular, we show that any ultracontractive semigroups always satisfy an exponential decay in the sense of 2.2. We treat different classes of examples, one of them containing the OrnsteinUhlenbeck-type semigroup and Γ∗-semigroup. We apply our results to fractional powers of non-negative self-adjoint semigroup. We derive a simple criterium on the function charaterizing the exponential decay to deduce ultracontractivity property and relations that must satisfy the ultracontractive bounds an heat kernel of the semigroup.