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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 $u\left(z\right)=u\left(\lambda z\right)$ for $\left|\lambda \right|=1$, $z\in \partial \Omega$ we construct a holomorphic function $f\in 𝕆\left(\Omega \right)$ such that $u\left(z\right)={\int }_{𝔻z}{\left|f\right|}^{p}d{𝔏}_{𝔻z}^2$ for $z\in \partial \Omega$, where $𝔻=\{\lambda \in ℂ|\lambda |<1\}$.

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 $E\left(f\right)=E$.

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 $\partial {𝔹}^n$ we give necessary and sufficient conditions in order that there exists a holomorphic function $f\in 𝕆\left({𝔹}^n\right)$ such that $$u\left(z\right)={\int }_{\left|\lambda \right|<1}{\left|f\left(\lambda z\right)\right|}^2\mathrm{d}{𝔏}^2\left(\lambda \right).$$