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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 $g(\cdot) \in C^{m}(\mathbb{R}^n)$.

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First, I think you need to replace $g(x,y)\leq 0$ by $g(x,y) < 0$; for example, with $g(x,y)\leq 0$, if you take $g(x,y)$ to be a plane in $\mathbb{R}^3$ crossing the $xy$-plane along the $x$-axis, with $g(x,y) > 0$ for all $y > 0$ and $g(x,y) < 0$ for $y < 0$, then you're in trouble: $F$ is not even continuous even for some analytic $f$ (notice that $g$ in this case is also analytic). Otherwise, the problem is reduced to smoothness of $\int_{h_1(y)}^{h_2(y)}f(x)d(x)$ as a function of $y$, provided smoothness of $h_i$. Now use the fundamental theorem (of course, you have to take care of some geometric conditions - i.e. investigate intersection of the surface $\{(x,y,g(x,y))\in \mathbb{R}^n\times\mathbb{R}\}$ with $\mathbb{R}^n\times\{0\}$, investigate smoothness of this intersection).

For instance, in case $g: \mathbb{R}^2\rightarrow\mathbb{R}$, transversality of intersection of the graph of $g$ with the $xy$-plane would ensure that the region $\{(x,y): g(x,y) < 0\}$ has smooth boundary, and the problem, as stated above, is reduced to investigating $\int_{h_1(y)}^{h_2(y)}f(x)d(x)$.

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