I'm wondering about the spatial smoothness of the heat kernel $K(t,x,x_0)$ on Lipschitz and polygon domains (or cornered domains). It's well known that $K(t,x,x_0)$ is smooth in $t$ for very general domains. How about the spatial regularity?
There are some results for the Laplace kernel on cornered domains using microlocal analysis (b-calculus), but I didn't find results for heat kernel.
You can definitely find some papers on coefficients of the trace of the heat kernel in this setting (e.g. https://arxiv.org/abs/0901.0019), but this might not be what you want.
Otherwise, I am not sure if there is a computation of the heat kernel performed in such generality, but I recall hearing that polygons fit within the framework of stratified spaces of depth 2, so they might be studied using an iterated version of edge calculus, a generalisation of b-calculus that follows the same philosophy, fitting into Melrose's (and others) program of Lie algebroids and groupoids. This framework provides a road map with which to compute a heat kernel, but it could be that this specific case has some particularities that make it more difficult.