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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Niro Yanagihara
Journal: Trans. Amer. Math. Soc. 180 (1973), 449-461
MSC: Primary 30A78
DOI: https://doi.org/10.1090/S0002-9947-1973-0338382-X
MathSciNet review: 0338382
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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 ${N^ + }$.


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Keywords: The class <!– MATH ${N^ + },{N^ + }$ –> <IMG WIDTH="78" HEIGHT="43" ALIGN="MIDDLE" BORDER="0" SRC="images/img8.gif" ALT="${N^ + },{N^ + }$">, zV as an <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$F$">-space in the sense of Banach, multiplier as a closed operator, local unboundedness of the space <IMG WIDTH="37" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img7.gif" ALT="${N^ + }$">, representations of continuous linear functionals on the space <IMG WIDTH="37" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${N^ + }$">
Article copyright: © Copyright 1973 American Mathematical Society