# Estimates of Dirichlet heat kernels for unimodal L\'evy processes with low intensity of small jumps

## TL;DR

This paper derives two-sided estimates for the transition densities of unimodal Lévy processes with low jump intensity, extending understanding of heat kernels in cases where the Lévy density's scaling index matches the Euclidean dimension.

## Contribution

It provides the first two-sided Dirichlet heat kernel estimates for Lévy processes with Lévy densities that have a weak lower scaling index equal to the Euclidean dimension.

## Key findings

- Establishes two-sided heat kernel estimates for a broad class of Lévy processes.
- Covers cases with Lévy densities that are regularly varying with index equal to the dimension.
- Extends previous results to processes with low jump intensity and matching scaling index.

## Abstract

In this paper, we study transition density functions for pure jump unimodal L\'evy processes killed upon leaving an open set $D$. Under some mild assumptions on the L\'evy density, we establish two-sided Dirichlet heat kernel estimates when the open set $D$ is $C^{1, 1}$. Our result covers the case that the L\'evy densities of unimodal L\'evy processes are regularly varying functions whose indices are equal to the Euclidean dimension. This is the first results on two-sided Dirichlet heat kernel estimates for L\'evy processes such that the weak lower scaling index of the L\'evy densities is not necessarily strictly bigger than the Euclidean dimension.