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

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.