The second Cousin problem with bounded data

Edgar Lee Stout

Vol. 26 (1968), No. 2, 379–387

DOI: 10.2140/pjm.1968.26.379

Abstract

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

Mathematical Subject Classification

Primary: 32.20

Milestones

Received: 6 April 1967

Published: 1 August 1968