Let $W^{1,2}(\mathbb D)$ be the complex valued Sobolev space on $\mathbb D$ where $\mathbb D $ is the open unit disk of the complex plane. By definition, $W^{1,2} (\mathbb D)$ is the set of all complex valued locally integrable functions in the open unit disc such that $f \in L^2 (\mathbb D)$ and the weak derivatives $\frac{\partial f}{\partial z}$ , $\frac{\partial f}{\partial \bar z}$ are both in $L^2 (\mathbb D)$.

Since the boundary of $\mathbb D$ is the unit circle $\mathbb T = \{z \in \mathbb C : \lvert z \rvert=1\}$, there is a trace operator $T : W^{1,2} (\mathbb D) \to L^2 (\mathbb T)$ which is bounded operator and $Tf = f|_{\mathbb T}$ whenever $f\in C(\overline {\mathbb{D}}) \cap W^{1,2} (\mathbb D)$.

Is it true that the Poisson integral of $Tf$ is in $W^{1,2} (\mathbb D)$ whenever $f \in W^{1,2} (\mathbb D)$? I suspect that this is true, however, I have not been able to prove this. Hints would be appreciated.

Are there any references which involve the study of properties of traces and Poisson integrals? Any suggestions would be appreciated!