Let $O$ be a non empty open subset of a bounded open set $\Omega\subset \mathbb{R}^n$ and let $f\in L^2(\Omega)\cap W^{2,\infty}(O)$. Let $u: \Omega \to \mathbb{R}$ be a function such that the function $g$ defined by $$g(x)=\begin{cases}u(x),\quad &x\in O \\ 0,\quad &x\in \Omega\setminus O\end{cases} $$ is an element of ${\cal C}^2(\overline{\Omega})$.
Let $h$ be defined by $$h(x)= \begin{cases} u(x) f(x), \quad &x\in O\\ 0,\quad & x\in \Omega\setminus O\end{cases} $$
Do we have $h \in W^{2,\infty}(\Omega)$?