Consider the following sequence of domains $\Omega_n = \mathbb R^d \setminus B(0,1/n)$ for $n \geq 1$.
For such domains, there exists total extension operators $$ E_n : W^{m,p}(\Omega_n) \rightarrow W^{m,p}(\mathbb R^d) $$ for $m \in \mathbb N_{\geq 1}$ and $1 \leq p < +\infty$.
My question is the following : can I find a family operator $(E_n)_{n \geq 1}$ for which the family of operator norm $(||E_n||)_{n \geq 1}$ ($m,p$ are fixed) is uniformly bounded ?
I was thinking that it might be possible because, heuristically, $E_{n}$ converges to the identity operator as $n \to +\infty$.
I was thinking of considering any extension operator $E_1$ on $\Omega_1$ and get the other $E_n$ by rescaling. I did some computations and it would work if the extension operator $$ \partial^{\alpha}E^1 = \sum_{|\gamma| = |\alpha|} E_{\gamma} \partial^{\gamma} $$ for some bounded operators $E_{\gamma}$. This is true for the half-space (see Adams, Sobolev spaces, Theorem 5.19), but I don't know if this would hold for my domain.
Otherwise, we get a factor at least $r^{-1}$ in the norm of $E_r$ and the scaling trick does not work. If this helps, we can restrict to radial Sobolev functions.