Heat kernel of anisotropic nonlocal operators

TL;DR

This paper constructs and estimates the fundamental solution for highly anisotropic, space-inhomogeneous integro-differential operators using the Levi method, with applications to related Cauchy problems.

Contribution

It introduces a novel approach to analyze anisotropic nonlocal operators and provides explicit estimates for their fundamental solutions.

Findings

Constructed fundamental solutions for anisotropic operators

Provided estimates for the solutions' behavior

Applied results to solve associated Cauchy problems

Abstract

We construct and estimate the fundamental solution of highly anisotropic space-inhomogeneous integro-differential operators. We use the Levi method. We give applications to the Cauchy problem for such operators.