  In this paper we derive estimates for the Hessian of the logarithm
(log-Hessian) for solutions to the heat equation. For initial data in the form
of log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian
admits an explicit, uniform (in space) lower bound. This yields a new estimate
for the Lipschitz constant of a transport map pushing forward the standard
Gaussian to a measure in this class. Further connections are discussed with
score-based diffusion models and improved Gaussian logarithmic Sobolev
inequalities. Finally, we show that assuming only fast decay of the tails of
the initial datum does not suffice to guarantee uniform log-Hessian upper
bounds.
