Modification of L. Hörmander's method in problems on generators and its applications (Q1916561)

Let \(\Omega\) be an open set in \(\mathbb{C}^n\), \(H(\Omega)\) be the space of holomorphic functions in \(\Omega\) provided with the standard topology of a Fréchet space, \(F\) be a fixed collection of functions \(f_j\in H(\Omega)\), \(1\leq j<N\), \(N\leq\infty\). Let, further, \(E,G_1,G_2,\dots\), be certain subspaces in \(H(\Omega)\) and \(G\) be a subspace in \(G_1\times G_2\times\cdots\). The author says that a pair \((F,G)\) generates \(E\) if every function \(\Phi\in E\) can be represented in the form \(\Phi=\sum_{1\leq j<N} f_jg_j\), where \((g_1,g_2,\dots)\in G\). A pair \((F,G)\) regularly generates \(E\) if in addition every such sum \(\sum f_jg_j\) is an element of \(E\). The paper is devoted to the study of generating and regularly generating pairs in inductive limits of weighted Banach spaces. It contains sufficient conditions for a given pair \((F,G)\) to generate or regularly generate \(E\) in the case when the spaces defining \(G\) and \(E\) are consisting of functions satisfying integral or uniform weight estimates. Also it contains some criteria for regularly generating pairs with slight auxiliary restrictions. The author obtains the results by help of his modification of known Hörmander's methods.