Suppose we have a domain $\Omega \subset \mathbb{R}^n$ and a specific function $f \in L^{k,p}(\Omega)$. Now suppose that one can prove that for the domain $\Omega$ in question it is impossible to construct an extension operator $\Lambda: L^{k,p}(\Omega) \rightarrow L^{k,p}(\mathbb{R}^n)$. Does it follow that $f \notin L^{k,p}(\mathbb{R}^n)$ ?
By an extension operator I mean an operator that maps $f$ to itself as in: $\Lambda f|_\Omega = f$. Basically the question is, is it possible to have a function $f \in L^{k,p}(\Omega)$ AND $f \in L^{k,p}(\mathbb{R}^n)$ while simultaneously being able to prove that it is impossible to construct the mapping $\Lambda: L^{k,p}(\Omega) \rightarrow L^{k,p}(\mathbb{R}^n)$ ? Or is the fact that $\Lambda$ does not exist enough to show that $f \notin L^{k,p}(\mathbb{R}^n)$ ?