I have a question regarding the definition of general trace spaces (e.g. from the book "Elliptic problems in domains with piecewise smooth boundaries" by Nazarov and Plamenevsky (Chapter 2.2.4)). Let $K=\{(x,y)\in \mathbb{R}^2 \,\mid\, r>0, \theta\in (0,\alpha)\}\subset \mathbb{R}^2$ be an angle of opening $\alpha\in(0,2\pi]$, where $(r,\theta)$ denote the polar coordinates of $(x,y)$. For $l\in \mathbb{N}_0$ and $\gamma\in \mathbb{R}$ define $V^{l}_\gamma(K):= \{u\colon K\to \mathbb{R} \,\mid\, ||u||_{l,\gamma, K}<\infty \}$ where $||u||_{l,\gamma, K}^2:= \sum_{|\alpha|\leq l}\int_K r^{2(\gamma-l+|\alpha|)}|D^\alpha u|^2 dxdy$. Then the space $V^{l+\tfrac 12}_\gamma(\partial K)$ is defined as the "space of traces on $\partial K$ of functions in $V^{l+1}_\gamma(K)$" and equipped with the norm $||u||_{l+\tfrac 12,\gamma, \partial K}:=\inf\{||U||_{l+1,\gamma, K}\mid U=u \text{ on } \partial K\}$.
I have some questions regarding this definition:
Question 1: What does "space of traces on $\partial K$ of functions in $V^{l+1}_\gamma(K)$" actrually mean? How can we define a trace of a $V^{l+1}_\gamma(K)$-function? As far as I know the idea for e.g. bounded sets with Lipschitz boundary is to construct a trace operator using density results. In this case, the boundedness is not given. And why is it true that $C^\infty(\overline{K})\cap V^{l+1}_\gamma(K)$ is dense in $V^{l+1}_\gamma(K)$?
Question 2: Is there a similar way to define trace spaces if we replace $K$ by a set which has no Lipschitz boundary? E.g. a domain with exterior cusp at the origin?