Multipliers and linear functionals for the class $N^{+}$

Abstract:

Multipliers for the classes ${H^p}$ are studied recently by several authors, see Duren’s book, Theory of ${H^p}$ spaces, Academic Press, New York, 1970. Here we consider corresponding problems for the class ${N^ + }$ of holomorphic functions in the unit disk such that \[ \lim \limits _{r \to 1} \int _0^{2\pi } {{{\log }^ + }} |f(r{e^{i\theta }})|d\theta = \int _0^{2\pi } {{{\log }^ + }|f({e^{i\theta }})|} d\theta < \infty .\] Our results are: 1. ${N^ + }$ is an $F$-space in the sense of Banach with the distance function \[ \rho (f,g) = \frac {1}{{2\pi }}\int _0^{2\pi } {\log (1 + |f({e^{i\theta }}) - g({e^{i\theta }})|)} d\theta .\] 2. A complex sequence $\Lambda = \{ {\lambda _n}\}$ is a multiplier for ${N^ + }$ into ${H^q}$ for a fixed $q,0 < q < \infty$, if and only if ${\lambda _n} = O(\exp [ - c\sqrt n ])$ for a positive constant $c$. 3. A continuous linear functional $\phi$ on the space ${N^ + }$ is represented by a holomorphic function $g(z) = \Sigma {b_n}{z^n}$ which satisfies ${b_n} = O(\exp [ - c\sqrt n ])$ for a positive constant $c$. Conversely, such a function $g(z) = \Sigma {b_n}{z^n}$ defines a continuous linear functional on the space ${N^ + }$.