Suppose that $u_j\rightharpoonup u$ in $W^{1,p}(\Omega)$ (notice the weak convergence), with $\Omega\subset \mathbb{R}^3$ regular enough. Let $v_j=Tu_j$, and $v=Tu$, where $T:W^{1,p}(\Omega)\to L^p(\partial \Omega)$ is the trace operator. Can I claim one of the following?
- $v_j\rightharpoonup v$ in $W^{1-\frac{1}{p},p}(\partial \Omega)$
- $v_j\to v$ in $L^p(\partial \Omega)$
In my case $p=4/3$, so I think the first implies the second. But what I really need is simply the second one.
Edit: the fractional exponent for the trace Sobole space is $1-\dfrac{1}{p}$, not $\dfrac{1}{2}$ (unless $p=2$, obviously).
Edit 2: I read somewhere (can't find the link/reference now) that a valid norm for for traces is
$$ \|u\|_{W^{1-1/p,p}(\partial \Omega)} := \underset{Tv=u}{\inf} \|v\|_{W^{1,p}(\Omega)}. $$ It looks to me like a legit norm, and with this characterization, we have that $T$, as an operator from $W^{1,p}(\Omega)$ onto $W^{1-1/p,p}(\partial\Omega)$ is continuous, with norm $1$. If this is correct, then, as Jose said in a comment, 1. would be true.