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Mapping properties of the Bergman projection on convex domains of finite type. (English) Zbl 0801.32008
This paper contains the definitive results on mapping properties of the Bergman projection on bounded convex domains of finite type with smooth boundary in \(\mathbb{C}^ n\). The authors prove that the Bergman projection is a bounded mapping from the Sobolev space \(L^ p_ k\) into itself for \(1< p<\infty\) and \(k\in \mathbb{N}\); from the standard Lipschitz space \(\Lambda_ \alpha\) into itself for \(0< \alpha<\infty\); and from \(L^ \infty\) into BMO.
They also show that the Bergman projection preserves suitably defined anisotropic Lipschitz spaces \(\Gamma_ \alpha\).
The proofs depend on geometric constructions and estimates for the Berman kernel function and its derivatives on convex domains of finite type that were obtained previously by the first author [‘Estimates on the Bergman kernels of convex domains’, Adv. Math., to appear].

MSC:
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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