On invariant domains of holomorphy

A. Sergeev

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In 1945 the first author introduced the classes $H^{p} (α)$ , 1 ≤ p<∞, α > -1, of holomorphic functions in the unit disk with finite integral (1) ∬ |f(ζ)|p (1-|ζ|²)α dξ dη < ∞ (ζ=ξ+iη) and established the following integral formula for $f \in H^{p} (α)$ : (2) f(z) = (α+1)/π ∬ f(ζ) ((1-|ζ|²)α)/((1-zζ̅)2+α) dξdη, z∈ . We have established that the analogues of the integral representation (2) hold for holomorphic functions in Ω from the classes $L^{p} (Ω; {[K (w)]}^{α} d m (w))$ , where: 1) $Ω = w = (w ₁, . . ., w_{n}) \in ℂ^{n} : I m w ₁ > \sum_{k = 2}^{n} | w_{k} | ²$ , $K (w) = I m w ₁ - \sum_{k = 2}^{n} | w_{k} | ²$ ; 2) Ω is the matrix domain consisting of those...

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Annales Polonici Mathematici

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A survey of properties of invariant pseudodistances and pseudometrics is given with special stress put on completeness and product property.

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Some remarks on holomorphic extension in infinite dimensions

Pham Ban (1994)

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In finite-dimensional complex analysis, the extension of holomorphic maps has been investigated by many authors. In recent years some authors have considered this problem in the infinite-dimensional case. The aim of the present note is to study the extension of holomorphic maps with values in some Banach complex manifolds.