I have a question about unbounded domains of $\mathbb{R}^d$.
In the following, $d \ge 2$ and $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connected open subset of $\mathbb{R}^d$.
Adams' book (enter link description here) say that there exists a total extension operator for $D$ if $D$ is a domain satisfying the strong local Lipschitz condition (see condition A). Roughly speaking, extension operator $E$ is a bounded linear operator from $W^{m,p}(D)$ to $W^{m,p}(\mathbb{R}^d)$ such that $Ef=f$ a.e. on $D$.
But this strong local Lipschitz condition is very complicated for me. I do not understand when this condition satisfied (in paticular, when $D$ is unbounded).
Condition(A): $D$ is minimally smooth domain (strong local Lipschitz condition). That is, there exist positive numbers $\delta$ and $M$, a locally finite open cover $\{U_j\}$ of boundary $\partial D$, and, for each $j$ a real-valued function $f_j$ of $d-1$ variables, such that the following conditions hold:
(i) For some finite $R$, every collection of $R+1$ of the sets $U_j$ has empty intersection.
(ii) For every pair of points $x,y \in D_{\delta}$ such that $|x-y|<\delta$, there exists $j$ such that $x,y \in \{z \in U_{j}:d(z ,\partial U_{j})>\delta\}$.
(iii) Each function $f_j$ satisfies a Lipschitz condition with $M$: that is, if $\xi=(\xi_{1},\ldots,\xi_{d-1})$ and $\eta=(\eta_{1},\ldots,\eta_{d-1})$ are in $\mathbb{R}^{d-1}$, then $|f(\xi)-f(\eta)|\le M|\xi-\eta|$.
(iv) For some Cartesian coordinate system $(\xi_{1,j},\ldots,\xi_{j,d})$ in $U_j$, $D \cap U_{j}$ is represented by the inequality $\xi_{j,d}<f_{j}(\xi_{j,1},\ldots,\xi_{j,d-1})$.
My question
Let us consider the following "good" condition:
Condition(B): $D$ is smooth domain. That is, for each $x \in \partial D$, there exist $r_x>0$ and a continuously differentiable function $F^x:\mathbb{R}^{d-1} \to \mathbb{R}$ such that (upon rotating and relabeling the coordinate axes if necessary) we have \begin{align*} D \cap Q(x,r_x)=\{y \in \mathbb{R}^d:F^x(y_{1},\ldots,y_{d-1})<y_{d}\} \cap Q(x,r_x), \end{align*} where $Q(x,r)=\{y \in \mathbb{R}^d : |y_{i}-x_{i}|<r, i=1,\ldots,d\}$.
If $D$ satisfies condition(B) then $D$ satisfies condition(A)?
Note If $D$ is bounded, condition(A) reduces to the simple condition that $D$ should have a locally Lipschitz boundary, that is, that each point $x \in \partial D$ should have a neighborhood $U_x$ whose intersection with $\partial D$ should be the graph of a Lipschitz continous function. Therefore, when $D$ is bounded, condition(B) implies condition(A).