It is well-known that if $\phi:\mathbb{R}^n \longrightarrow \mathbb{R}$ is a function with $\phi(0) = 0$, $\phi(x)>0$ if $x \neq 0$ and $D^2\phi(0) \geq 0$, then the integral $$\int_{\mathbb{R}^n} e^{-\phi(x)/2t}a(x)\,\mathrm{d}x$$ has an asymptotic expansion as $t\rightarrow 0$ (obtained via the Laplace method). One can also calculate asymptotic expansions under other non-degeneracy conditions than $D^2\phi(0) \geq 0$ or in the case that the zero set of $\phi$ is a submanifold.
\Edit: I assume that $\phi$ and $a$ are smooth and that $a$ is compactly supported. However, if there are results/references where this is not the case, I would be interested in this case as well.
Question: What if the critical set is arbitrary?
- How bad can the critical set actually be?
- Under which assumptions on the critical set can we calculate (at least first order) asymptotic expansions? What kind of irregular sets can be taken here?
