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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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