I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature.

For each $s>0$, I have a solution $\{u_s\}$ of the conjugate heat equation $\Box ^* u_s = 0$ with some 'nice' initial data.

My question is: does anyone have good sources for results like "If we have such-and-such an estimate on the $\{u_s\}$, then we get convergence $\{u_s\} \to u$ smoothly on compact subsets, where $u$ is the fundamental solution of the conjugate heat equation."

It is proving exceedingly difficult to find good parabolic regularity results written down cleanly somewhere!