Consider a cylinder $C_{R,T} = D_R \times (0,T)$ in $\mathbb{R}^n$, where $D_R$ is a ball of radius $R$ in $\mathbb{R}^{n-1}$ and where $x$ parametrizes the ball $D_R$ and $t$ parametrizes the interval $(0,T)$. Then the Sobolev trace theorem says that we can define the trace or boundary value at $t=0$ of each function $f \in W^{1,p}(C_{R,T})$ as some function $Tf \in W^{1-1/p,p}( D_R)$.
In motivating the result, one can easily show using the Fundamental Theorem of Calculus (see e.g. Giusti, Direct Methods in the Calculus of Variations, p.106-107) that if $f$ is smooth in $C_{R,T}$ and if $\{t_n\} \to 0$, the trace $Tf$ is the limit in $L^p(D_R)$ of the functions $f_n(x) \equiv f(t_n,x)$.
My question: Is it also true that $Tf$ is the limit of the $f_n$ in $W^{1-1/p,p}(D_R)$? After all, by the trace theorem they all belong to $W^{1-1/p,p}(D_R)$.