Fix $k\in\mathbb Z_+$ and $1<p<\infty$, for $U\subset\mathbb R^n$ (w.l.o.g $\partial U\in C^\infty$) , extension theorem shows that there is a bounded linear operator $E_U:W^{k,p}(U)\to W^{k,p}(\mathbb R^n)$. But $\|E_U\|_{U\to\mathbb R^n}$ can be highly depend on $U$.
Denote $T_m=\{x=(x',x_n)\in\mathbb R^n:|x'|<2^{-m},\frac12<x_n<\frac52\}$, can we find a series of extension $E_m:W^{n,2}(T_m)\to W^{n,2}(\mathbb R^n)$ that is uniformly bounded for all $m$? (That what I want to prove mostly)
In general, is there any good characterization of domains such that the extension operators can be uniformly bounded. That is, can we find a "good" class of domains (subset of $\mathbb R^n$) $\mathscr F$, such that, there is extension operators $\{E_U:W^{k,p}(U)\to W^{k,p}(\mathbb R^n)\}_{U\in\mathscr F}$ and $\|E_U\|\le C(k,p,\mathscr F)$.
My motivation comes from Sobolev Space using Fourier transform. Fractional Sobolev Space defined as $W^{s,p}(U)=\{f\in L^p(U):\exists\tilde f\in L^p(\mathbb R^n),\tilde f|_U=f,\mathcal F^{-1}[(1+|\xi|)^s\mathcal F(\tilde f)]\in L^p(\mathbb R^n)$, with norm $\|f\|_{W^{s,p}}'=\inf_{\tilde f|_U=f}\|\tilde f\|_{W^{s,p}}$.
Can prove $\|f\|_{W^{k,p}(U)}'\approx_{k,p,U}\|f\|_{W^{k,p}(U)}$ when $k$ is positive integer. But when can we put away the dependence of $U$.