It is well known that for $p\in(1,\infty)$ and any $f\in L^p(\mathbb{T})$ function $Cf(z)=\int\limits_{\mathbb{T}}\frac{f(\zeta)}{1-\bar\zeta z}dm(\zeta)$ (where $m$ is normalized Lebesgue measure on unit circle) is holomorphic in $\mathbb{D}$, $~g=\lim\limits_{r\to 1-} Cf(re^{it})$ is finite a.e. and $\|g\|_{L^p}\leq C_p \|f\|_{L^p}$. I try to understand is it true for multi-dimensional case. It easy to see that for $f\in L^p(\mathbb{T}^n)$ function $\int\limits_{\mathbb{T}^n}\frac{f(\zeta)dm(\zeta)}{(1-\bar\zeta_1z_1)...(1-\bar\zeta_n z_n)}$ is holomorphic in $\mathbb{D}^n$. I suppose that estimation of norm should follows from one-dimensional case but how can I derive it? I have tried to find something about boundness of Cauchy transform in polydisc but I failed.
Thanks in advance for any help.