Optimal Weighted \texorpdfstring{$L^2$}{L2} Hessian Estimates for Parabolic Equations: General Diffusion and Coefficient Mismatch

Abstract

We study weighted $L^2$ Hessian estimates under diffusion marginals, allowing the equation's principal matrix to be chosen independently. For regular uniformly elliptic coefficients, bounded reference drift and compactly supported doubling initial laws, the optimal large-damping constant on the full interval is the maximum of initial, flat and propagation contributions. Finiteness yields weighted Sobolev well-posedness and a contraction criterion for nonlinear perturbations. An explicit operator formula determines the initial contribution; propagation bounds become exact under uniform directional limits or asymptotic isotropy at infinity. Under geometric assumptions including nonnegative Ricci curvature, propagation is determined by terminal momenta of action-minimizing paths. For reference covariance $a$, matching principal matrix $a/2$ and Hessian normalization $a^{1/2}D^2u\,a^{1/2}$, every initial law gives a finite estimate. Without an additional time weight, the limiting constant lies between $2$ and $2\sqrt2$: finite atomic laws attain the upper bound, while nondegenerate Gaussian laws and their finite mixtures attain the lower bound.