A remark on the uniform extendability of the Bergman kernel function.(English)Zbl 0756.32008

Let $$D$$ be a smoothly bounded domain in $$\mathbb{C}^ n$$, $$H^ 2(D)$$ the space of $$L^ 2$$-integrable holomorphic functions on $$D$$, and $$H(\overline D)$$ that of holomorphic functions which can be extended holomorphically to some open set containing $$\overline D$$. Take definition: condition $$Q$$ holds on $$D$$ if $$P$$ maps $$C_ 0^ \infty(D)$$ into $$H(\overline D)$$, where $$P$$ is the orthogonal projection from $$L^ 2(D)$$ onto $$H^ 2(D)$$. In this note, the author proves that condition $$Q$$ holds on $$D$$ is equivalent to the following statement: For each compact subset $$K$$ of $$D$$, there exists an open set $$D_ K$$ containing $$\overline D$$ such that the Bergman kernel function $$K(z,w)$$ associated to $$D$$ can be extended holomorphically in $$z$$ and anti-holomorphically in $$w$$ to $$K(z,w)\in C^ \omega(D_ K\times D)$$. As a corollary, condition $$Q$$ implies an extension version of H. Cartan’s theorem. And strictly pseudoconvex domains, Reinhardt domains and circular domains picking up some transverse symmetries on the boundary all satisfy the condition $$Q$$.

MSC:

 32D15 Continuation of analytic objects in several complex variables 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) 32T99 Pseudoconvex domains 32A07 Special domains in $${\mathbb C}^n$$ (Reinhardt, Hartogs, circular, tube) (MSC2010)
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