We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball: For a function $u$ which is lower semi-continuous on $\partial \mathbb{B}^{n}$ we give necessary and sufficient conditions in order that there exists a holomorphic function $f\in \mathbb{O}(\mathbb{B}^{n})$ such that \[ u(z)=\int _{|\lambda |<1}\left|f(\lambda z)\right|^{2}\mathrm{d}{\mathfrak L}^{2}(\lambda ).\].
We solve the following Dirichlet problem on the bounded balanced domain $\Omega $ with some additional properties: For $p>0$ and a positive lower semi-continuous function $u$ on $\partial \Omega $ with $u(z)=u(\lambda z)$ for $|\lambda |=1$, $z\in \partial \Omega $ we construct a holomorphic function $f\in \Bbb O(\Omega )$ such that $u(z)=\int _{\Bbb Dz}|f|^pd \frak L_{\Bbb Dz}^2$ for $z\in \partial \Omega $, where $\Bbb D=\{\lambda \in \Bbb C\:|\lambda |<1\}$.
For $ z\in \partial B^n$, the boundary of the unit ball in $\mathbb{C}^n$, let $\Lambda (z)=\lbrace \lambda \:|\lambda |\le 1\rbrace $. If $ f\in \mathbb{O}(B^n)$ then we call $E(f)=\lbrace z\in \partial B^n\:\int _{\Lambda (z)}|f(z)|^2\mathrm{d}\Lambda (z)=\infty \rbrace $ 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 $\mathbb{P}^{n-1}=\mathbb{P}(\mathbb{C}^n)$ then there exists a holomorphic function $f$ in the unit ball $B^n$ so that $E(f)=E$.