Super-Poincar\'e and Nash-type inequalities for Subordinated Semigroups

TL;DR

This paper establishes that super-Poincaré and Nash-type inequalities for a symmetric semigroup generator imply similar inequalities for functions of the generator, such as fractional powers and logarithms, generalizing previous results.

Contribution

It proves the transfer of super-Poincaré and Nash inequalities through Bernstein functions of the generator, extending known results to a broader class of functions.

Findings

Super-Poincaré inequalities imply similar inequalities for $-g(A)$ with Bernstein functions g.

Results apply to fractional powers and logarithmic functions of the generator.

Provides multiple examples illustrating the generalized inequalities.

Abstract

We prove that if a super-Poincar\'e inequality is satisfied by an infinitesimal generator $-A$ of a symmetric contracting semigroup then it implies a corresponding super-Poincar\'e inequality for $-g(A)$ with any Bernstein function $g$. We also study the converse statement. We deduce similar results for the Nash-type inequality. Our results applied to fractional powers of $A$ and to $\log(I+A)$ and thus generalize some results of Biroli and Maheux, and Wang 2007. We provide several examples.