Levi extension theorems for meromorphic functions of weak type in infinite dimension (Q504580)

The authors study Levi type extension problems for meromorphic functions with values in a topological vector space. Let \(F\) be a sequentially complete locally convex space and let \(W\subset F'\) be such that a set \(B\subset F\) is bounded iff \(u(B)\) is bounded for every \(u\in W\). Let \(\Delta\subset\mathbb C\) denote the unit disc and let \(\Delta_r:=r\Delta\). Let \(D\subset\mathbb C^n\) be open and let \(D_\ast\subset D\) be dense in \(D\). Assume that \(f\) is a meromorphic \(F\)-valued function on \(D\times(\Delta_r\setminus\overline\Delta)\) (\(r>1\)) such that for each \(z\in D_\ast\) there exists a set \(P_z\subset\Delta_r\) without accumulation points in \(\Delta_r\) such that for every \(u\in W\) the function \(u(f(z,\cdot))\) has a meromorphic extension to \(\Delta_r\) whose pole set is contained in \(P_z\) and which is locally bounded on \(\Delta_r\setminus P_z\). Then \(f\) extends meromorphically to \(D\times\Delta_r\). The authors discuss also generalizations of the above theorem under weaker assumptions on \(F\) and \(W\).