ACTA issues

Topologies on some subclasses of the Smirnov class

Romeo Meštrović, Žarko Pavićević

Acta Sci. Math. (Szeged) 69:1-2(2003), 99-108

Abstract. For $p>1$, the class $N^p$, introduced by I. I. Privalov with the notation $A_q$ in [11], is defined as the space of analytic functions $f$ on the open unit disk $D$ in ${\bf C}$ for which $(\log ^ +| f(z)| )^p$ has a harmonic majorant on $D$. However, the space $N^p$ can be expressed as a union of certain weighted $H^2$ spaces and it is given a locally convex topology ${\cal H}_p$ as the inductive limit of these spaces. We note that this topology coincides with the Mackey topology (strongest locally convex topology yielding the same dual) on $N^p$. We then consider the individual spaces $H^2(w)$, where $w$ is a positive function on the unit circle such that $w$ and $| \log w| ^p$ are summable. Our results are in fact generalizations of those obtained by J. E. McCarthy in [8] for the case $p=1$. In particular, we give asymptotic versions of Szegő's theorem and the Helson--Szegő theorem on these spaces, and characterize the universal multipliers of their duals.

AMS Subject Classification (1991): 30D55, 46J15

Received June 25, 2001. (Registered under 2886/2009.)