Heat kernel upper bounds for symmetric Markov semigroups

TL;DR

This paper establishes that Nash-type inequalities and heat kernel upper bounds for symmetric Markov semigroups are equivalent, extending previous results to more general cases using a new generalized Davies' method.

Contribution

It introduces a generalized Davies' method to prove the equivalence between Nash inequalities and off-diagonal heat kernel bounds for symmetric Markov semigroups, broadening prior results.

Findings

Nash inequalities imply off-diagonal heat kernel bounds.

Off-diagonal heat kernel bounds imply Nash inequalities.

Results extend previous work to more general settings.

Abstract

It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of \cite{CKS} for Nash-type inequalities with power order considerably and also extend that of \cite{Gri} for second order differential operators on a complete non-compact manifold.