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Let $M$ be a smooth manifold with Riemannian metric $g$, which is not smooth but only continuous.

Question: Is there still an asymptotic expansion of the heat kernel of the form $$ p_t(x, y) \sim (4 \pi t)^{-n/2} e^{-\frac{1}{4t}d(x, y)^2}\sum_{j=0}^\infty t^j \Phi_j(x, y)$$ where the correction terms $\Phi_j$ are only continuous (or some other regularity that makes sense)?

Assume the following "mild form" of irregularity of $g$: Assume $g$ is smooth everywhere except at a submanifold $N$, where it is still smooth in direction of $N$ and only non-smooth in directions transversal to $N$, but maybe still Lipschitz-continuous.

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The Varadhan's asymptotics

$t \ln p_t(x,y) \to_{t \to 0} -\frac{d^2(x,y)}{4}$

is known to hold in Lipschitz Riemannian manifolds. It was proved by James Norris in his paper:

Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds, Acta Mathematica, Volume 179, Issue 1, pp 79-103

This Varadhan's asymptotics was later generalized by K.T. Sturm, [J. Math. Pures Appl. 75 (1996), 273–297] and Ramirez [Comm. Pure Appl. Math. 54 (2001), 259–293] to even more singular spaces.

Concerning the full Minakshisundaram–Pleijel expansion for the heat kernel on singular spaces, I am not aware of general results. This is still an active domain of research. There is an interesting example that I like, that comes from a measurable Riemannian structure on the Sierpinski gasket. See for instance the short summary

Measurable Riemannian structure and its heat kernel analysis on the Sierpinski gasket

It shows that weird and interesting things can happen.

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