Poisson kernels and pluriharmonic ${\cal H}^2$ functions on homogeneous Siegel domains

Bartosz Trojan

Abstract: In the paper we prove that a real function $F$ defined on a homogeneous not necessarily symmetric Siegel domain satisfying an ${cal H}^2$ condition is pluriharmonic if and only if ${\bf H} F=0$, ${\cal L}F=0$, $L F=0$, where ${\bf H}$, $\cal L$, $L$ are second order differential operators. This generalizes the result of Damek, Hulanicki, Müller, and Peloso [3] where symmetric domains were considered. Our approach to study non-symmetric case is based on $T$-algebras introduced by Vinberg [11].

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