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Weighted integral representations of pluriharmonic functions in the Siegel domain of \(C^n\). (English) Zbl 1430.32008

Summary: The Siegel domain in the space \(C^n\) is defined as follows: \[\Omega_n= \left\{\eta =(\eta_1,\eta_2,\ldots ,\eta_n)\in C^n: \mathrm{Im\,} \eta_1>\sum_{k=2}^n |\eta_k|^2\right \}. \] In the paper the weighted spaces \(L_{\alpha }^p(\Omega_n)\) with the weight function of the type \((\mathrm{Im\,}\eta_1-\sum_{k=2}^n |\eta_k|^2)^{\alpha }\) 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.

MSC:

32Q02 Special domains (Reinhardt, Hartogs, circular, tube, etc.) in \(\mathbb{C}^n\) and complex manifolds
32U05 Plurisubharmonic functions and generalizations
31C10 Pluriharmonic and plurisubharmonic functions
32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
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