Functional Inequalities and Subordination: Stability of Nash and Poincar\'e inequalities

We show that certain functional inequalities, e.g.\ Nash-type and Poincar\'e-type inequalities, for infinitesimal generators of $C_0$ semigroups are preserved under subordination in the sense of Bochner. Our result improves \cite[Theorem 1.3]{BM} by A.\ Bendikov and P.\ Maheux for fractional powers, and it also holds for non-symmetric settings. As an application, we will derive hypercontractivity, supercontractivity and ultracontractivity of subordinate semigroups.