Partial higher regularity

[pabryan]

In section 6 should be outline (or just mention) that partial higher regularity may be obtained on open sets where the metric is more regular? That is, we should still have smoothing effects of the heat kernel dependent on the regularity of the heat kernel.

[lashputin]

I know how to do this if we have a globally comparable metric: in this case, we switch to a divergence form operator globally, and then where we have higher regularity, it becomes non-divergence form, and then we can do Schauder theory. But if this wasn't the case, I can't see how to do it. I mean, the point is that when we localise our estimates, even though we start in the domain of an operator (globally) in L^2, our localisation is just distributional, and we can obtain harnack there. Only if we had global comparability can i see that we can obtain this still in the domain of an operator inside the localisation.