Two-sided Gaussian estimates for fundamental solutions of second-order parabolic equations in non-divergence form

TL;DR

This paper proves two-sided Gaussian bounds for fundamental solutions of second-order parabolic equations in non-divergence form, using minimal regularity assumptions and avoiding complex previous methods.

Contribution

It provides a simpler proof of Gaussian estimates for equations with Dini mean oscillation coefficients, relying on local boundedness and weak Harnack inequality.

Findings

Established two-sided Gaussian bounds under minimal regularity.

Simplified proof avoiding normalized adjoint solutions.

Applicable to equations with Dini mean oscillation coefficients.

Abstract

We establish two-sided Gaussian bounds for the fundamental solution of second-order parabolic operators in non-divergence form under minimal regularity assumptions. Specifically, we show that the upper and lower bounds follow from the local boundedness property and the weak Harnack inequality for the adjoint operator $P^*$, respectively. This provides a simpler and more direct proof of the Gaussian estimates when the coefficients have Dini mean oscillation in $x$, avoiding the use of normalized adjoint solutions required in previous works.