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

2025/05/19 by Kim, Seick, Lee, Sungjin, Sakellaris, Georgios
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.12659

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.

Citations

• Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients
• Estimates for fundamental solutions of parabolic equations in non-divergence form
• Doubling properties for second order parabolic equations
• A new proof of Moser's parabolic harnack inequality using the old ideas of Nash

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