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Reproducing properties and \(L^ p\)-estimates for Bergman projections in Siegel domains of type II. (English) Zbl 0842.32016
The authors study \(L^p\)-mapping properties of weighted Bergman projections on Siegel domains of type II. Let \(L^{p, \varepsilon}\) denote the space of functions that are \(p\)th power integrable with respect to Lebesgue measure weighted by \(B(z,z)^{-\varepsilon}\), where \(B\) denotes the ordinary Bergman kernel function, and let \(P_\varepsilon\) denote the orthogonal projection from \(L^{2,\varepsilon}\) onto its holomorphic subspace. (The authors determine necessary and sufficient conditions on \(\varepsilon\) for this holomorphic subspace to be nontrivial.) After proving a Plancherel-Gindikin formula for the holomorphic subspace of \(L^{2,\varepsilon}\), the authors derive relations among \(\varepsilon\), \(p\), and \(r\) for \(P_\varepsilon\) to be bounded on \(L^{p,r}\). The paper is based on the second author’s 1993 thesis.

MSC:
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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