Let $k(x)$ be a function with lower bound $k_0\leq k(x)$ over the domain $\Omega\subset \mathbb{R}^2$.
Let us define the weighted norm $$ \|u\|_{1,k,\Omega}= \int_{\Omega} k(x) u^2dx+\int_{\Omega}k(x)\nabla u\cdot\nabla u dx. $$
In the Hilbert space $H^1(\Omega)$, if $u\in H ^1(\Omega)$, we already know that $\text{tr } u \in H^{\frac{1}{2}}(\partial\Omega)$.
Is there something similar in our weighted space? If $u\in H^1_k(\Omega)$, do we have $\text{tr } u\in H^\frac{1}{2}_k(\partial\Omega) $?
Moreover, if I want to define the dual space of $H^{\frac{1}{2}}_k(\partial\Omega)$, could the norm be
$$\|w\|_{H^{-\frac 1 2}_k(\partial\Omega)}=\sup_{u\in H^1_k(\Omega)} \frac{\langle u, w\rangle}{\|u\|_{H^1_k(\Omega)}}$$?