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

Seick Kim, Sungjin Lee, Georgios Sakellaris

TL;DR

The paper addresses obtaining two-sided Gaussian bounds for the fundamental solution $\Gamma(t,x,s,y)$ of a second-order parabolic operator in non-divergence form under minimal regularity. The main approach shows that the upper bound follows from the local boundedness property of the adjoint $P^*$ and the lower bound from the weak Harnack inequality for $P^*$; under the Dini mean oscillation in $x$ condition on $A$, these properties hold, yielding the standard Gaussian bounds $\frac{1}{N_1 (t-s)^{d/2}} \exp\{-\kappa_1 \frac{|x-y|^2}{t-s}\} \le \Gamma \le \frac{N_1}{(t-s)^{d/2}} \exp\{-\frac{|x-y|^2}{\kappa_1 (t-s)}\}$. This provides a direct route to Gaussian bounds without relying on normalized adjoint solutions and aligns with, yet generalizes beyond, prior results under $ ext{DMO}_x$ (e.g., DEK21). The work thus connects Harnack-type properties of $P^*$ to sharp heat-kernel-type estimates for $P$, extending classical results to non-divergence form parabolic equations with minimal regularity.

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.

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

TL;DR

The paper addresses obtaining two-sided Gaussian bounds for the fundamental solution of a second-order parabolic operator in non-divergence form under minimal regularity. The main approach shows that the upper bound follows from the local boundedness property of the adjoint and the lower bound from the weak Harnack inequality for ; under the Dini mean oscillation in condition on , these properties hold, yielding the standard Gaussian bounds . This provides a direct route to Gaussian bounds without relying on normalized adjoint solutions and aligns with, yet generalizes beyond, prior results under (e.g., DEK21). The work thus connects Harnack-type properties of to sharp heat-kernel-type estimates for , extending classical results to non-divergence form parabolic equations with minimal regularity.

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 , respectively. This provides a simpler and more direct proof of the Gaussian estimates when the coefficients have Dini mean oscillation in , avoiding the use of normalized adjoint solutions required in previous works.
Paper Structure (3 sections, 6 theorems, 50 equations)

This paper contains 3 sections, 6 theorems, 50 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['thm1']}
  3. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1.3

Suppose that the formal adjoint operator $P^*$ satisfies the local boundedness property. Given $T>0$, let $N_0$ be the constant associated with $R_0=\sqrt{T}$ in Definition def_lb. Then the fundamental solution $\Gamma(t,x,s,y)$ of $P$ satisfies the upper Gaussian bound: for any $x$,$y\in \mathbb{R} where $\kappa=\kappa(\Lambda)>0$ and $N=N(d,\lambda, \Lambda, N_0)>0$.

Theorems & Definitions (11)

  • Definition 1.1: Local Boundedness Property
  • Definition 1.2: Weak Harnack Inequality
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6: Dong--Escauriaza--Kim DEK21
  • Lemma 2.1
  • proof
  • Lemma 3.1: Safonov--Yuan SY99
  • Lemma 3.2
  • ...and 1 more