Meromorphic Extensions of (·,W)-Meromorphic Functions

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 C n , in the sense u ∘ f is meromorphic for all u ∈ W , is meromorphic. Further, combining those results with studing on ( BB )-Zorn property we give conditions for Fréchet 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 ∩ 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 ( · , W ) -meromorphic function on a Riemann domain D over a Fréchet space coincide with the envelope of holomorphy of D ? (2) When will ( · , W ) -meromorphic functions be able to extend meromorphically through an analytic subset of codimension ≥ 2 of a domain in a Fréchet space?