On the topological description of weighted inductive limits of spaces of holomorphic and harmonic functions (Q1293182)

For a bounded open set \(G\) in \(\mathbb{C}\) and a decreasing sequence \(V= (v_n)_{n\in\mathbb{N}}\) of continuous functions of \(\overline G\), strictly positive on \(G\), denote by \(Hv_n(G)\) the Banach space of all holomorphic functions on \(G\) for which \(\|f\|_n:= \sup_{z\in G}|f(z)|v_n(z)< \infty\) and let \(VH(G)\) be the inductive limit of the spaces \(Hv_n(G)\). Furthermore, let \(\overline V\) be the set of all upper semicontinuous functions \(\overline v:G\to [0,\infty[\) for which \({v\over v_n}\) is bounded on \(G\) for each \(n\in\mathbb{N}\) and let \[ H\overline V(G)= \Biggl\{f\in H(G):\|f\|_{\overline v}:= \sup_{z\in G}|f(z)|\overline v(z)< \infty \forall\overline v\in\overline V\Biggr\}. \] The authors derive a necessary condition for the topological equality \(VH(G)= H\overline V(G)\) in terms of the behaviour of \(V\) on \(\partial\overline G\). They use it to generate out of non-distinguished Köthe sequence spaces \(\lambda_1(A)\) examples of weights \(V\) on the unit disk \(\mathbb{D}\) for which \(VH(G)\neq H\overline V(G)\) topologically, thereby simplifying earlier examples of \textit{J. Bonet} and \textit{J. Taskinen} [Mich. Math. J. 42, 259-268 (1995; Zbl 0841.46014)] and \textit{J. Bonet} and \textit{S. N. Melikhov} [J. Math. Anal. Appl. 205, No. 2, 454-460 (1997)]. If the holomorphic functions are replaced by harmonic functions in the above then the arguments even give a characterization of \(h\overline V(G)= Vh(G)\) under mild assumptions on \(V\).