Weak Poincar\'e inequalities in the absence of spectral gaps

TL;DR

This paper investigates weak Poincaré inequalities for Markov semigroup generators without spectral gaps, linking spectral density bounds to decay rates, with applications to pseudodifferential operators, heat semigroups, and fractional Laplacians.

Contribution

It establishes a connection between spectral density near zero and weak Poincaré inequalities, extending the understanding to pseudodifferential operators and classical inequalities.

Findings

Derived bounds on density of states lead to weak Poincaré inequalities.

Revealed optimal decay rates for heat and fractional Laplacian semigroups.

Unified classical Nash inequality as a special case of WPI.

Abstract

For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called "weak Poincar\'e inequality" (WPI), originally introduced by Liggett [Ann. Probab., 1991]. Applications to general classes of constant coefficient pseudodifferential operators are studied. Particular examples are the heat semigroup and the semigroup generated by the fractional Laplacian in the whole space, where the optimal decay rates are recovered. Moreover, the classical Nash inequality appears as a special case of the WPI for the heat semigroup.