Pointwise estimates for the ground states of some classes of positivity preserving operators

TL;DR

This paper derives pointwise estimates for ground states of certain positivity preserving operators, specifically negatively perturbed local Dirichlet operators, using Green's kernel, with implications for heat kernel estimates.

Contribution

It provides new pointwise bounds for ground states of perturbed Dirichlet operators and proves the existence of their Green's kernels, enhancing understanding of their spectral properties.

Findings

Establishment of pointwise estimates for ground states

Proof of Green's kernel existence for these operators

Sharp estimates that recover known results

Abstract

We establish pointwise estimates for the ground states of some classes of posi- tivity preserving operators. The considered operators are negatively perturbed (by measures) strongly local Dirichlet operators. These estimates will be written in terms of the Green's kernel of the considered operators, whose existence will be proved. In many circumstances our estimates are even sharp so that they recover known results about the subject. The results will deserve to obtain large time heat kernel estimates for the related operators.