Weighted Nash Inequalities

TL;DR

This paper introduces a versatile method using weighted Nash inequalities to derive non-uniform bounds on kernel densities of Markov semigroups, extending classical results and providing new tools for analyzing heat kernels.

Contribution

The work presents a simple, general approach based on weighted Nash inequalities to obtain bounds on kernel densities, applicable to a wide class of Markov processes.

Findings

Derived non-uniform bounds on heat kernel densities

Controlled trace and Hilbert-Schmidt norms of heat kernels

Applied method to heat kernels associated with exponential-type measures

Abstract

Nash or Sobolev inequalities are known to be equivalent to ultracontractive properties of Markov semigroups, hence to uniform bounds on their kernel densities. In this work we present a simple and extremely general method, based on weighted Nash inequalities, to obtain non-uniform bounds on the kernel densities. Such bounds imply a control on the trace or the Hilbert-Schmidt norm of the heat kernels. We illustrate the method on the heat kernel on $\dR$ naturally associated with the measure with density $C_a\exp(-|x|^a)$, with $1