Holomorphic functions on balls in an almost complex abstract Wiener space
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.
\(L^ 2\)-converging Taylor series expansion
scientific article; zbMATH DE number 894103