Suppose $\Omega$ is a sufficiently smooth bounded domain of $\mathbb{R}^2$. Let $f \in H^2(\Omega) = W^{2,2}(\Omega)$ and $g \in H^2(\Omega) \cap L^{\infty}(\Omega)$.
Can we say that $g \circ f \in H^2(\Omega)$?
I tried looking at other questions here but they only have composition with smooth or Lipschitz functions.
If $g(0)=0$, it is enough to have $g\in H^2_{\text{loc}}(\mathbb{R})$. This is a theorem due to Bourdau. See Theorem 12.72 in Sobolev book. If $g(0)\ne 0$ and $g\in H^2_{\text{loc}}(\mathbb{R})$, define $h(t)=g(t)-g(0)$. Then $h\circ f \in H^2(\Omega)$. In turn, so does $g\circ f=h\circ f+g(0)$ since constants are in $H^2$ in bounded domains.