The sheaf of outer functions in the polydisc

Abstract

where m is the Haar measure on the unit circle T, k is an absolutely integrable real-valued function on T, and λ is a complex number of modulus one. Closely related to the class of outer functions is the Smirnov class JV*(Z7), which consists of all functions that are holomorphic in U and admit an inner-outer factorization. The class N*(U) is an algebra, and the outer functions are precisely the invertible elements of this algebra. An alternative characterization of N*(U), considered by Rudin in [5], where it was extended to the polydisc U, is that a holomorphic function f in U belongs to N*(U) if and only if there exists a strongly convex function φ (depending on /) for which ^(Log |/|) has a harmonic majorant. This definition can be extended naturally to arbitrary polydomains WΊ x W2 x ••• x Wn, the requirement now being that ^(Log + |/|) have an ^-harmonic majorant in W1 x W2 x x Wn. We define the outer functions in W1 x W2 x x Wn to be the invertible elements of the algebra N^(W1 x W2 x x Wn). (For the polydisc U, this definition can easily be seen to agree with the one given by Rudin in [5, Def. 4.4.3, p. 72].) The correspondence that assigns to each polydomain W in C the group O( W Π U) of outer functions in the intersection W Π U*f defines a sheaf <g? on the closure U of U, which is locally determined in the sense that Γ(U, &) is canonically isomorphic to the group of outer functions in U. Our aim, in this article, is to show that the cohomology groups H(U, &) are trivial for all integers ? ^ 1. Sheaves of a similar type (sheaves of germs of holomorphic functions satisfying boundary conditions on polydomains) have been studied by Nagel in [4], where a unified approach to many types of