Meromorphic Extensions of $$(\cdot , W)$$-Meromorphic Functions

Abstract

In this paper we establish some classes of subspaces W of the dual $$F'$$ of a locally convex space F such that every F-valued (F, W)-meromorphic function (with/without local boundedness) on a domain D in $$\mathbb {C}^n,$$ in the sense $$u \circ f$$ is meromorphic for all $$u \in W,$$ is meromorphic. Further, combining those results with studing on (BB)-Zorn property we give conditions for Frechet spaces E, F and subspaces W of $$F'$$ under which (F, W)-meromorphic functions can be meromorphically extended to a domain D of E from a subset $$D \cap E_B$$ where $$E_B$$ is the linear hull of some balanced convex compact subset B of E. Using these results we get the answers of the following questions: (1) When does the domain of meromorphy of a $$(\cdot , W)$$ -meromorphic function on a Riemann domain D over a Frechet space coincide with the envelope of holomorphy of D? (2) When will $$(\cdot , W)$$ -meromorphic functions be able to extend meromorphically through an analytic subset of codimension $$\ge 2$$ of a domain in a Frechet space?