Non-commutative Nash inequalities

TL;DR

This paper extends Nash inequalities to non-commutative Lp spaces in quantum Markov processes, establishing their relationship with Poincare and log-Sobolev inequalities and proving them for certain semigroups.

Contribution

It introduces non-commutative Nash inequalities and explores their connections to other functional inequalities in quantum settings.

Findings

Nash inequalities are established for unital reversible semigroups.

The relationship between Nash, Poincare, and log-Sobolev inequalities is clarified.

Theoretical framework extends classical inequalities to quantum Markov processes.

Abstract

A set of functional inequalities - called Nash inequalities - are introduced and analyzed in the context of quantum Markov process mixing. The basic theory of Nash inequalities is extended to the setting of non-commutative Lp spaces, where their relationship to Poincare and log-Sobolev inequalities are fleshed out. We prove Nash inequalities for a number of unital reversible semigroups.