Estimates for fundamental solutions of parabolic equations in non-divergence form

TL;DR

This paper constructs fundamental solutions for second order parabolic equations in non-divergence form with Dini mean oscillation coefficients, establishing sub-Gaussian and Gaussian bounds under various regularity conditions.

Contribution

It introduces a method to construct fundamental solutions under Dini mean oscillation assumptions and proves Gaussian bounds for these solutions.

Findings

Fundamental solutions satisfy sub-Gaussian estimates.

Gaussian bounds hold when coefficients are Dini continuous in space.

Method applies to second order parabolic systems in non-divergence form.

Abstract

We construct the fundamental solution of second order parabolic equations in non-divergence form under the assumption that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case when the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish the Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form.