The property \((\tilde \Omega)\) and holomorphic functions (Q1384888)

Let \(E\) be a nuclear Fréchet space with a fundamental system of seminorms \(\| \cdot \|_k\). \(E\) is said to have property \((\widetilde\Omega)\) if \[ \forall p \exists q, d>0 / \forall k \exists c>0 : \| \cdot \|_q ^{*1+d} \leq c \| \cdot \|_k ^* \| \cdot \|_p ^{*d}, \] and property \(({\overline{\overline\Omega}})\) if \[ \forall p \exists q / \forall d>0 \forall k \exists c>0 : \| \cdot \|_q ^{*1+d} \leq c \| \cdot \|_k ^* \| \cdot \|_p ^{*d}. \] Also, a holomorphic function \(f:E\longrightarrow F\) is said to be of uniform type if it factors through the canonical map \(E\longrightarrow E_p\), for some continuous seminorm \(p\). In this paper the author gives a series of equivalences relating properties \((\widetilde\Omega)\), \(({\overline{\overline\Omega}})\), the uniform type of some functions, and the existence of Dirichlet representation for others. In particular, \(E\) has property \(({\overline{\overline\Omega}})\) if and only if every holomorphic function \(\Lambda_1 ^* (\alpha)\longrightarrow E^*\) is of uniform type for all \(\alpha = (\alpha_j)\) for which \[ \Lambda_1 (\alpha)=\{ (z_j): \sum_{j\geq 1} | z_j| r^{\alpha_j}<\infty, 0\leq r <1\} \] is nuclear.