Weakly holomorphic extensions and the condition \((L)\) (Q1281600)

Let \(A\) be a subset of a locally convex complex vector space \(E\). We say that \(A\) satisfies condition (\( L_o\)) at \(a\in A\) if for every sequence of complex-valued continuous polynomials \(\{P_k\}\) on \(E\) with \(|P_k(x)|\leq M(x)<+\infty\), \(k\in \mathbb N\), \(x\in A\), and for every \(\varepsilon>0\) there exist \(C>0\) and a neighborhood \(\Omega\) of \(a\) such that \(|P_k(x)|\leq C(1+\varepsilon)^{\deg P_k}\), \(x \in \Omega\), \(k\geq 1\). One says that \(A\) satisfies the condition \((L)\) at \(a\) if \(A\cap \Omega\) satisfies \((L_o)\) at \(a\) for every neighborhood \(\Omega\) of \(a\). The authors study relations between the condition \(L\) and weakly holomorphic extensions and separate analyticity in infinite dimension. In particular, they obtain the following improvements of \textit{Nguyen Thanh Van} [Ann. Polon. Math. 33, 71-83 (1976; Zbl 0336.46050)]. Let \(E\) and \(F\) be complex Frechet spaces. Let \(U\subset E\), \(V\subset F\) be open sets. Let \(A\) be a subset of \(U\) such that \(A\) satisfies \(L\) at some point \(a \in A\). Then the following statements are true. \(1^o\). Let \(f:A\to F\) be a map such that for all \(u\in F'\) the function \(uf\) has a holomorphic extension to \(U\). Then \(f\) admits a holomorphic extension to \(U\). \(2^o\). Let \(f:U\times V \to \mathbb C\) be a function such that \(f(\cdot,w)\) is holomorphic on \(U\) for every \(w \in V\), and \(f(z, \cdot)\) is holomorphic on \(V\) for every \(z\in A\). Then \(f\) is holomorphic on \(U\times V\).