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Associated weights and spaces of holomorphic functions. (English) Zbl 0934.46027
Let $$G$$ 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.

##### 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 (should also be assigned at least one classification number from Section 30-XX) 46A04 Locally convex Fréchet spaces and (DF)-spaces
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