Gaussian Estimates: A Brief History

Abstract: Two-sided Gaussian estimates for the fundamental solution of a second order linear parabolic differential equation are upper and lower bounds in terms of the fundamental solution of the classical heat conduction equation. In his seminal 1958 paper Nash stated, without proof, two-sided non-Gaussian bounds for the fundamental solution of a uniformly parabolic divergence structure equation assuming only boundedness of the coefficients. In his 1967-1968 papers Aronson derived truly Gaussian estimates for the fundamental solutions of a large class of linear parabolic equations (including the divergence structure equation) under minimal non-regularity assumptions on the coefficients. Subsequently in 1986 Fabes & Stroock derived Gaussian estimates for the divergence structure equation directly from the ideas of Nash and went on to prove Nash's continuity theorem and the Harnack inequality as a consequence of their estimate. In this note I describe these results together with various extensions.