Duality of holomorphic functions spaces und smoothing properties of the Bergman projection

Details Duality of holomorphic functions spaces und smoothing properties of the Bergman projection Duality of holomorphic functions spaces und smoothing properties of the Bergman projection

2011-10-07

arxiv.org/abs/1110.1533v1

Mathematics

Complex Variables

Scientific paper

Let $\Omega\subset\mathbb{C}^n$ be a bounded domain with smooth boundary, whose Bergman projection $B$ maps the Sobolev space $H^{k_{1}}(\Omega)$ (continuously) into $H^{k_{2}}(\Omega)$. We establish two smoothing results: (i) the full Sobolev norm $\|Bf\|_{k_{2}}$ is controlled by $L^2$ derivatives of $f$ taken along a single, distinguished direction (of order $\leq k_{1}$), and (ii) the projection of a conjugate holomorphic function in $L^{2}(\Omega)$ is automatically in $H^{k_{2}}(\Omega)$. There are obvious corollaries for when $B$ is globally regular.

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