Given a complex manifold
M, an open covering 𝒱 ≡{Vα}α∈A, and, for each α ∈ A a function fα
holomorphic on Vα such that for all α, α′∈ A, fαfα′−1 is a zero-free holomorphic
function on Vα∩ Vα′, the associated second Cousin problem is the problem of
showing the existence of a function F holomorphic on M such that for all
α, Ffα−1 is a zero free function holomorphic on Vα. In the present paper we
consider an analogous problem in the case that M is the open unit polycylinder
UN= {(z1,⋯,zN) ∈CN: |z1| < 1,⋯,|zN| < 1}, that the functions fα are
required to be bounded and that the sought function F is also required to be
bounded.
