Moduli of bounded holomorphic functions in the ball

We prove that there is a continuous non-negative function g on the unit sphere in C d, d ≥ 2, whose logarithm is integrable with respect to Lebesgue measure, and which vanishes at only one point, but such that no non-zero bounded analytic function m in the unit ball, with boundary values m⋆, has |m⋆| ≤ g almost everywhere. The proof analyzes the common range of co-analytic Toeplitz operators in the Hardy space of the ball. 0 Introduction Let Bd be the unit ball in C , Sd be the boundary of Bd, and σd be normalized Lebesgue measure on Sd. Every function m in H (Bd), the space of bounded analytic functions in Bd, has radial limits σd-almost everywhere on Sd, defining a function m ⋆ on the sphere; conversely, m can be recovered from m by integrating against the Szegö kernel. The problem which this paper addresses is when a given non-negative bounded function g on Sd is the modulus |m| of some function m in H(Bd). When d = 1, the problem was completely solved by Szegö [10]: a necessary and sufficient condition that g be the modulus of a non-zero function in H(B1) is