Weighted Integral Representations of Pluriharmonic Functions in the Siegel Domain of $$C^n$$

Abstract

The Siegel domain in the space $$C^n$$ is defined as follows: $$\begin{aligned} \Omega _n=\left\{ \eta =(\eta _1,\eta _2,\dots ,\eta _n)\in C^n: Im \eta _1>\sum _{k=2}^n |\eta _k|^2 \right\} . \end{aligned}$$In the paper the weighted spaces $$L_{\alpha }^p(\Omega _n)$$ with the weight function of the type $$(Im\eta _1-\sum _{k=2}^n |\eta _k|^2)^{\alpha }$$ are introduced. For pluriharmonic functions u from the spaces $$L_{\alpha }^p(\Omega _n)$$ weighted integral representations are established. Inequalities between weighted $$L^p$$-norms of conjugate pluriharmonic functions are shown.