Super Poincaré and Nash-type inequalities for Subordinated Semigroups
نویسندگان
چکیده
We prove that if a super-Poincaré inequality is satisfied by an infinitesimal generator −A of a symmetric contraction semigroup on L2 and that is contracting on L1, then it implies a corresponding super-Poincaré inequality for −g(A) for any Bernstein function g. We also study the converse of this statement. We prove similar results for Nash-type inequalities. We apply our results to Euclidean, Riemannian, hypoelliptic and OrnsteinUhlenbeck settings.
منابع مشابه
Logarithmic Sobolev inequalities and Nash-type inequalities for sub-markovian symmetric semigroups
1 We study relationships between Logarithmic Sobolev inequalities with one parameter of Davies-Simon type, energy-entropy inequality, Nash-type inequality and Sobolev-type inequalities. The inequalities of Sobolev-type apply in the general setting of symmetric sub-Markovian semigroups (and some generalizations). We provide several examples of application of theses results for ultracontractive s...
متن کاملDimension-independent Harnack Inequalities for Subordinated Semigroups
Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the BakryEmery curvature condition, the subordinate semigroup with power α satisfies a dimension-free Harnack inequality provided α ∈ ` 1 2 , 1 ́ , and it satisfies the log-Harnack inequality for all α ∈ (0, 1). Some infinite-dimensional examples are also presen...
متن کاملOn Equivalence of Super Log Sobolev and Nash Type Inequalities
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 bor...
متن کاملNash-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 functi...
متن کاملSecond order Poincaré inequalities and CLTs on Wiener space
We prove in nite-dimensional second order Poincaré inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian elds, Stein's method and Malliavin calculus. We provide two applications: (i) to a new second order characterization of CLTs on a xed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated elds.
متن کامل