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.