Let Ω be a domain of finite type in ℂ² and let f be a function holomorphic in Ω and belonging to ${}^{k,\alpha }\left(\Omega ̅\right)$. We prove the existence of boundary values for some suitable derivatives of f of order greater than k. The gain of derivatives holds in the complex-tangential direction and it is precisely related to the geometry of ∂Ω. Then we prove a property of non-isotropic Hölder regularity for these boundary values. This generalizes some results given by J. Bruna and J. M. Ortega for the unit ball.

For $z\in \partial B^n$, the boundary of the unit ball in ${ℂ}^n$, let $\Lambda \left(z\right)=\left\{\lambda \right|\lambda |\le 1\}$. If $f\in 𝕆\left(B^n\right)$ then we call $E\left(f\right)=\{z\in \partial B^n{\int }_{\Lambda \left(z\right)}|f\left(z\right){|}^2\mathrm{d}\Lambda \left(z\right)=\infty \}$ the exceptional set for $f$. In this note we give a tool for describing such sets. Moreover we prove that if $E$ is a $G_{\delta }$ and $F_{\sigma }$ subset of the projective $(n-1)$-dimensional space ${\mathbb{P}}^{n-1}=\mathbb{P}\left({ℂ}^n\right)$ then there exists a holomorphic function $f$ in the unit ball $B^n$ so that $E\left(f\right)=E$.