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Integral representations in general weighted Bergman spaces. (English) Zbl 1089.30037

This article concerns holomorphic functions in the upper half-plane that are \(p\)th-power integrable (where \(1\leq p<\infty\)) with respect to a positive, continuous weight function that depends only on the distance from the boundary. The weight function is allowed to approach \(0\) at the boundary at an arbitrary rate. Using a fractional integration technique, the author constructs an integral representation of Bergman type for such holomorphic functions. When \(p=2\), the author exhibits an isometric isomorphism between \(L^2({\mathbb R})\) and the space of weighted square-integrable holomorphic functions in the upper half-plane.

MSC:

30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
30C40 Kernel functions in one complex variable and applications
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