Associated weights and spaces of holomorphic functions.

*(English)*Zbl 0934.46027Let \(G\) be an open subset in \({\mathbb C}^N, N\geq 1\) and \(w\) a continuous positive function (a growth condition) on \(G\). Set \(B_w=\{f\in H(G):|f|\leq w\) on \(G\}\) and let \(\widetilde w(z)=\sup\{|f(z)|:f\in B_w\}\) be the associated with \(w\) weight function on \(G\). Several properties of weighted spaces of holomorphic functions on \(G\) are characterized in terms of associated weights \(\widetilde w\) in this interesting paper. As an example, it is shown that for a weight \(v:G\to{\mathbb R}_+\) on \(G\) with \(w=1/v\) and \(\widetilde v=1/\widetilde w\), the weighted Banach space of holomorphic functions \(Hv(G)=\{f\in H(G):\|f\|=\|f\|_v=\sup_Gv|f|<\infty\}\) is isomorphic to the corresponding space \(H\widetilde v(G)\). Further, it is shown that the natural biduality \(Hv_0(G)''=Hv(G)\), where \(Hv_0(G)=\{f\in H(G): vf\) tends uniformly to 0 as \(z\) approaches the boundary of \(G\}\), is an isometry if and only if \(\widetilde v=\widetilde v_0\). Given a decreasing sequence of weights \({\mathcal V}=\{v_n\}_n\) on \(G\), let \({\mathcal V}H(G)\) be the corresponding weighted (LB)-space of holomorphic functions associated with \({\mathcal V}\). Namely, \({\mathcal V}H(G)\) is the locally convex inductive limit \({\mathcal V}H(G)=\) ind\(_nHv_n(G)\). Let \(H{\overline V}(G)=\{f\in H(G): \|f\|_{{\overline v}}<\infty\) for every \({\overline v}\in{\overline V}\}\), where \({\overline V}={\overline V}({\mathcal V})\) is the set of all weights \({\overline v}\) on \(G\), for which \({\overline v}/v_n\) is bounded on \(G\) for every \(n\in{\mathbb N}\), be the projectve hull of \({\mathcal V}H(G)\).

The authors obtain characterizations of the bounded (DFS)-property and the bounded retractivity property for \({\mathcal V}H(G)\), and of the semi-Montel property for \(H\overline V(G)\) by inequalities involving associated weights. Particular examples of weighted spaces and their properties are provided and discussed. The question of when \({\mathcal V}H(G)=H{\widetilde V}(G)\) holds in topological sense for a naturally related with \(\{{\widetilde v}_n\}\) system \(\widetilde V\), is raised and discussed. By extending the use of their method, the authors also determine when two weighted topologies on the unit ball of a weighted Banach space of holomorphic functions coincide and when an embedding of such two spaces is compact. Finally, estimates for the associate weights \({\widetilde w}\) of radial growth conditions \(w\) on \({\mathbb C}\) or \(D\) under certain natural conditions are obtained.

The authors obtain characterizations of the bounded (DFS)-property and the bounded retractivity property for \({\mathcal V}H(G)\), and of the semi-Montel property for \(H\overline V(G)\) by inequalities involving associated weights. Particular examples of weighted spaces and their properties are provided and discussed. The question of when \({\mathcal V}H(G)=H{\widetilde V}(G)\) holds in topological sense for a naturally related with \(\{{\widetilde v}_n\}\) system \(\widetilde V\), is raised and discussed. By extending the use of their method, the authors also determine when two weighted topologies on the unit ball of a weighted Banach space of holomorphic functions coincide and when an embedding of such two spaces is compact. Finally, estimates for the associate weights \({\widetilde w}\) of radial growth conditions \(w\) on \({\mathbb C}\) or \(D\) under certain natural conditions are obtained.

Reviewer: T.V.Tonev (Missoula)

##### MSC:

46E10 | Topological linear spaces of continuous, differentiable or analytic functions |

46A11 | Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) |

46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |

46M40 | Inductive and projective limits in functional analysis |

30D15 | Special classes of entire functions of one complex variable and growth estimates |

30D55 | \(H^p\)-classes (MSC2000) |

32A22 | Nevanlinna theory; growth estimates; other inequalities of several complex variables |

32A30 | Other generalizations of function theory of one complex variable |

46A04 | Locally convex Fréchet spaces and (DF)-spaces |