Some characterizations of the properties \(\widetilde {LB}^\infty\) and \(LB_\infty\)
Let \(E\) be a Fréchet space with the structure generated by an increasing sequence \((\| \;\| _k)_{k=1}^\infty\) of seminorms. Put \(U_k:=\{x\in E: \| x\| _k<1\}\), \(\| u\| _k^\ast:=\sup\{| u(x)| : x\in U_k\}\), \(u\in E^\ast:=\) the strong dual of \(E\). We say that \(E\in LB_\infty\) if \[ \forall_{d_k\nearrow+\infty}\;\exists_p\;\forall_q\;\exists_{k(q)\geq q,\,C(q)>0}\;\forall_{x\in E}\;\exists_{m:q\leq m\leq k(q)}: \| x\| _q^{1+d_m}\leq C(q)\| x\| _m\| x\| _p^{d_m}. \] We say that \(E\in\widetilde{LB}^\infty\) if \[ \exists_{d_k\nearrow+\infty}\;\forall_p\;\exists_{q,\,k_0}\;\forall_{k\geq k_0}\;\exists_{C(k)>0}\;\forall_{u\in E^\ast}: \| u\| _q^{\ast\,1+d_k}\leq C(k)\| u\| _k^\ast\| u\| _p^{\ast\,d_k}. \] We say that a function \(f:D\rightarrow\mathbb C\) has a local Dirichlet representation if for every point \(x_0\in D\) there exist a neighborhood \(U\) and sequences \((\xi_k)_{k=1}^\infty\subset\mathbb C\), \((u_k)_{k=1}^\infty\subset E^\ast\) such that \(f(x)=\sum_{k\geq1}\xi_k\exp(u_k(x))\), \(x\in U\), and \(\sum_{k\geq1}| \xi_k| \exp(\| u_k\| ^\ast_K)<+\infty\) for every compact \(K\subset U\). The author presents various characterizations of \(\widetilde{LB}^\infty\) and \(LB_\infty\) spaces. In particular, he proves the following theorems: \(E\in\widetilde{LB}^\infty\) iff \[ \exists_{d_k\nearrow+\infty}\;\forall_p\;\exists_{q,\,k_0}\;\forall_{k\geq k_0}\;\exists_{C(k)>0}\;\forall_{r>0}: U_q\subset r^{d_k}U_k+\frac{C(k)}{r}U_p. \] Assume that \(E\in\widetilde{LB}^\infty\), let \(F\in LB_\infty\) be a Fréchet-Schwartz space, and let \(D\subset F^\ast\) be open. Then every holomorphic function \(f:D\rightarrow E^\ast\) is locally bounded. Let \(F\) be a nuclear Fréchet space. Then the following conditions are equivalent: (a) \(F\in LB_\infty\); (b) every separately holomorphic function on an open set \(U\times V\) of \(E\times F^\ast\), where \(E\in\widetilde{LB}^\infty\) is a nuclear Fréchet space having a basis, is holomorphic; (c) every separately holomorphic function as in (b) has a local Dirichlet representation.
scientific article; zbMATH DE number 2130969