Consider a smooth function $f(x)$ on $\mathbb{R}^{n}$ from some smoothness class $S_1$ and define $$ F(y) = \int\limits_{g(x,y)\leqslant 0}f(x)dx $$ where $y \in \Omega$, $\Omega$ is some domain in $\mathbb{R}^{k}$, $g(x,y)$ is some smoth function from smoothness class $S_{2}$. Is there some general way to obtain smoothness properties of function $F$?
For example, $S_1 = \mathcal{S}(\mathbb{R}^n)$ (Schwartz space), $g(x,y) = g(x*y)$, where $*$ is an addition or entrywise multiplication and $g(\cdot) \in C^{m}(\mathbb{R}^n)$.