Super Poincaré and Nash-type inequalities for Subordinated Semigroups
Résumé
We prove that if super Poincaré inequality is satisfied by an infinitesimal generator $-A$ of a symmetric sub-markovian semigroup then it implies a corresponding super Poincaré inequality for $-g(A)$ with any Bernstein function $g$. We also study the converse statement. We deduce similar results when the assumption of super Poincaré inequality is changed by a Nash-type inequality. In particular, we prove that if $D$ is a Nash function for $A$ then $g\circ D$ is essentially a Nash function for $g(A)$. Our results apply to fractional powers of $A$ and $\log(I+A)$ generalizing results of \cite{bm} and \cite{w1}. We provide several examples and settings of applications.
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