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The Szegö projection on convex domains. (English) Zbl 0948.32004
From the introduction: The purpose of this paper is to study the Szegö projection operator for a smoothly bounded convex domain \(\Omega\) of finite type in \(\mathbb{C}^n\). The results obtained are two-fold. First, the (essentially sharp) differential inequalities satisfied by the Szegö kernel are established. Second, sharp Sobolev space and Lipschitz space estimates for the projection operator are proved. The function space estimates in question are both isotropic (for the usual \(L^p_k(b\Omega)\) Sobolev spaces and the classical \(\Lambda_\alpha (b\Omega)\) Lipschitz spaces), and non-isotropic, i.e., for certain \(\Gamma_\alpha (b\Omega)\) Lipschitz spaces defined in terms of the geometry of the boundary of \(\Omega\).

MSC:
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32V35 Finite-type conditions on CR manifolds
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