This is only a too **long comment** trying to make sense of some obscurities/omissions in [Zobin's article](https://www.sciencedirect.com/science/article/pii/S0001870897916856). On page 99 there he writes $\bigcap_{\,l=1}^{\,k}W^l_\infty(\Omega)\not\subset L^\infty(\Omega)$ which suggest that he *does not* intend $W^l_\infty(\Omega)$ to be the usual Banach space $W^{l,+\infty}(\Omega)$ but instead a Fréchet spece in whose definition $C^k(\Omega)$ indeed is intended to be the space of *not necessarily bounded* functions with topology of uniform convergence of derivatines on compact sets. Then his application of the Open Mapping Theorem may be the following:

For any fixed $x\in\Omega$ define $f_x$ by $\Omega\owns y\mapsto d_{\,\Omega}(x,y)\,$. Then let $U$ in the space $E=W^1_\infty(\mathbb R^n)$ be the set of all $f_1$ with first order (a.e. defined weak) partial derivatives having absolute value less than $1$ on $\mathbb R^n$. Now $U$ is an open zero neighbourhood in $E$ whence by the Open Mapping Theorem there is a zero neighbourhood $V$ in $F=W^1_\infty(\Omega)$ with $V\subseteq R\,[\,U\,]\,$. Then there is a compact set $K\subseteq\Omega$ and $\varepsilon\in\mathbb R^+$ such that for $g$ in $F$ with $|\,g\,z\,|\le\varepsilon$ for $z\in K$ and first order partials having absolute value at most $\varepsilon$ on $\Omega$ we have $g\in V$.

Fixing some $x_0\in\Omega\,$, for any $x\,,z\in\Omega$ we have $f_x\,z\le f_{x_0}\,z+f_{x_0}\,x\,$. Since $f_{x_0}$ can be extended to a continuous function on $\mathbb R^n$, there is $M_9\in\mathbb R^+$ such that $f_x\,z\le M_9$ holds for all $x\,,z\in\Omega\,$. Taking $C=\varepsilon^{-1\,}(1+M_9)\,$, for  $g=C^{-1}f_x$ we see that $g\in V$ holds, and so there is $f_1\in U$ with $g=R\,f_1\,$. For $\tilde f=C\,f_1$ then $f_x\subseteq\tilde f$ with $|\,\partial^{\,\alpha}\tilde f\,z\,|< C$ for $z\in\mathbb R^n$ and $|\,\alpha\,|=1\,$.

Note that, if my interpretation above of Zobin's quite weird definition of the space $W^1_\infty(\Omega)$ as a topological linear supremum of the Fréchet space $C\,(\Omega)$ embedded in $\mathscr D'(\Omega)$ and the inverse image of the weak gradient from $L^{+\infty}(\Omega)$ is correct, then the condition $W^1_\infty(\Omega)\subset W^1_\infty(\mathbb R^n)\,|_\Omega$ is, at least apriori, much more restrictive than it would be for the usual definition of the Sobolev space $W^{1\,,+\infty}(\Omega)$ as a Banach space.