Holomorphic functions on balls in an almost complex abstract Wiener space
type: Item
instance of: scholarly article
title: Holomorphic functions on balls in an almost complex abstract Wiener space
zbMATH Open document ID: 0866.46027
author: Setsuo Taniguchi
published in: Journal of the Mathematical Society of Japan
publication date: 1997-07-31T00:00:00Z
review text: Two kinds of holomorphic functions on open balls of abstract Wiener spaces are introduced and, under certain geometrical conditions on the norm of the Wiener space, the coincidence of both notions is proved. Recall that an almost complex abstract Wiener space is a quadruple \((B,H,\mu,J)\), where \((B,H,\mu)\) is an abstract Wiener space and where \(J:B\to B\) is an isomorphism on both \(B\) and \(H\), satisfying the equation \(J^2=-Id\). Following the ideas of the Malliavin calculus it is possible to define now the Cauchy-Riemann operator \(\overline{\partial}\) for functions \(u\in L^2(B_R,d\mu)\) defined on the ball \(B_R=\{z\in B:|z|_B<R\}\). The first definition of holomorphy on \(B_R\) is now obtained by \[ \text{Hol}(B_R)= \{u\in L^2(B_R,d\mu)\mid \overline{\partial}u=0 \text{ on }B_R\}. \] The second definition of holomorphy comes from the approximation by polynomials. Let \({\mathcal P}_h(B)\) be the set of all holomorphic polynomials \({\mathcal P}(z)=\sum a_{k_1\dots k_n}\langle z,\ell_1\rangle^{k_1}\cdots\langle z,\ell_n\rangle^{k_n}\) on \(B\), where the \(\ell_j\)'s satisfy \(\langle Jz,\ell\rangle= i\langle z,\ell\rangle\), and let \({\mathcal H}^2(B_R)\) be the \(L^2(B_R,d\mu)\)-closure of \({\mathcal P}_h(B)\). Obviously, \({\mathcal H}^2(B_R)\subseteq \text{Hol}(B_R)\), and the author is interested in the converse inclusion. First of all, he proves Theorem 1. \(\text{Hol} (B_R)|_{B_r} \subseteq{\mathcal H}^2(B_r)\) for all \(r<R\), provided that the norm \(|\cdot|_B\) is \(J\)-rotationally invariant, i.e. \(|e^{\varphi J}z|_B=|z|_B\) for all \(z\in B\) and all \(0\leq\varphi< 2\pi\). Under the assumption of the theorem the author proves a mean value theorem for functions \(u\in\text{Hol} (B_R)\), and this leads to Theorem 2. For completely rotationally invariant norms \(|\cdot|_B\) and all \(R>0\) we have \(\text{Hol}(B_R)={\mathcal H}^2(B_R)\). Additionally, each \(u\in\text{Hol} (B_R)\) admits an \(L^2\)-converging Taylor series expansion.
reviewed by: Heinz Junek
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 894103
zbMATH Keywords: open balls of abstract Wiener spaces
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: Cauchy-Riemann operator
zbMATH Keywords: rotationally invariant norms
zbMATH Keywords: \(L^ 2\)-converging Taylor series expansion
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04740655
OpenAlex ID: W2086259938
DOI: 10.2969/JMSJ/04740655
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description: scientific article; zbMATH DE number 894103