I have a function $f : \mathbb R^2\to\mathbb C$ which is locally $H^1$, that is, $f|R\in H^1(R)$ for any bounded rectangle $R$. After a possible translation of $f$ I can assume that $f$ is absolutely continuous on each of the four lines $x=0$, $x=1$, $y=0$, and $y=1$. In particular, $f|_\Gamma$ is absolutely continuous on the boundary of the unit square $Q = [0,1]\times [0,1]$.
Now, let us consider the trace $T_\Gamma f$ of $f$ onto $\Gamma$. Then (as far as I know) some representative of $T_\Gamma f$ in $L^2(\Gamma)$ is $\tfrac 1 2$-Hölder continuous. Is it now true that $T_\Gamma f = f|_\Gamma$?