Trace of a function and $L^\infty$ norm

Let $\Omega$ be an open bounded domain in $\mathbb R^n$ and let $C= \lbrace(x, y): x\in\Omega, y\in [0, +\infty)\rbrace$.Let $$X(C)=\left\lbrace u\in L^2(C): u|_{\partial\Omega\times [0, +\infty)}=0 \ \text{ and } \int_C |\nabla u|^2 dx dy <+\infty\right\rbrace$$.

For $u\in C(\Omega)$ define$T:X(C)\to L^2(\Omega)$ be such that $T(u)=u(\cdot,0)$.

Assume that there is $k>0$ such that $||u||_{L^\infty(C)}\le k $.

Can we conclude that $||T(u)||_{L^\infty(\Omega)}\le k$?

I am interested in this because I am reading a proof in which it is stated that $u\in \mathcal C^{\alpha}(C)$ implies that $T(u)\in \mathcal C^\alpha(\Omega)$, so I am trying to get this through the norms, starting with the $L^\infty$ one.