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

TL;DR

This paper proves that key functional inequalities like Nash and Poincaré are preserved under subordination of semigroups, extending previous results and applying to non-symmetric cases, with implications for hypercontractivity properties.

Contribution

It extends the stability of Nash and Poincaré inequalities under subordination to non-symmetric semigroups, improving prior results for fractional powers.

Findings

Functional inequalities are preserved under subordination.

Results apply to non-symmetric semigroups.

Derived hypercontractivity, supercontractivity, and ultracontractivity.

Abstract

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.