Moduli of bounded holomorphic functions in the ball

B. Korenblum; J. McCarthy

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Moduli of bounded holomorphic functions in the ball

B. Korenblum·J. McCarthy

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Abstract

We prove that there is a continuous non-negative function $g$ on the unit sphere in $\cd$, $d \geq 2$, 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 $m^\star$, has $|m^\star| \leq g$ almost everywhere. The proof analyzes the common range of co-analytic Toeplitz operators in the Hardy space of the ball.

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B. Korenblum, J. McCarthy. 1993-09-30. Moduli of bounded holomorphic functions in the ball. https://arxiv.org/abs/math/9309202

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