On weighted spaces of holomorphic functions of several variables
Let \(\Omega\) be the complex \(d\)-space \({\mathbb C}^d\) or its open unit ball \({\mathbb B}_d\). For a weight function \(v\) on \(\Omega\), the authors consider the Banach space \(Hv(\Omega):=H^\infty_v(\Omega)\) consisting of all holomorphic functions \(f\) on \(\Omega\) for which the weighted sup-norm \(\|f\|_v:=\sup_{z\in\Omega} |f(z)|v(z)\) is finite. By a weight \(v\) on \(\Omega\) the authors mean a continuous radial function \(v: \Omega\to (0, \infty)\) such that \(\lim_{|z|\to R} |z|^mv(z)=0\) for all \(m\geq 0\) where \(R=\infty\) for \(\Omega={\mathbb C}^d\) and \(R=1\) for \(\Omega={\mathbb B}_d\). The main result of this paper is to obtain a complete description of conditions on \(v\) for \(Hv(\Omega)\) to be isomorphic to the sequence space \(\ell_\infty\), nontrivially extending earlier one-variable results by the first-named author [Stud. Math. 175, No. 1, 19--45 (2006; Zbl 1114.46020)] and the authors [J. Funct. Spaces Appl. 6, No. 1, 59--70 (2008; Zbl 1158.46017)]. For that purpose the authors also consider the little-oh subspaces \((Hv)_0(\Omega)\) and \(c_0\) of \(Hv(\Omega)\) and \(\ell_\infty\), respectively, and show that the following three assertions are equivalent: (i) \(v\) satisfies Condition (B); (ii) \(Hv(\Omega)\) is isomorphic to \(\ell_\infty\); (iii) \((Hv)_0(\Omega)\) is isomorphic to \(c_0\). Most part of the paper is devoted to establish the implication (i)\(\Longrightarrow\)(iii), which is the hard part of the characterization. Condition (B) is too technical to describe here, but it is satisfied (in case of the ball) by various standard weights. Moreover, the authors observe the polydisc pathology asserting that the space \(Hv({\mathbb D}^d) ((Hv)_0({\mathbb D}^d), resp.)\), similarly defined on the unit polydisc \({\mathbb D}^d\) with \(d\geq 2\), is never isomorphic to \(\ell_\infty\) (\(c_0\), resp.).
scientific article; zbMATH DE number 5728354
scientific article