THE REAL PART OF AN OUTER FUNCTION AND A HELSON-SZEG¨O WEIGHT

Abstract

Suppose F is a nonzero function in the Hardy space H1. We study the set ff ; f is outer and jF j · Re f a.e. on @Dg, where @D is the unit circle. When F is a strongly outer function in H1 and ° is a positive constant, we describe the set ff ; f is outer, jF j · ° Re f and jFi 1j · ° Re (f i 1) a.e. on @Dg. Suppose W is a Helson-Szeg¨o weight. As an application, we parametrize real-valued functions v in L1(@D) such that the difference between logW and the harmonic conjugate function ~v of v belongs to L1(@D) and kvk1 is strictly less than ¼=2 using a contractive function ® in H1 such that (1 + ® )=(1 i ®) is equal to the Herglotz integral of W.