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 | References | Similar Articles | Additional Information

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 ${N^ + }$.

- Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523** - P. L. Duren,
*On the multipliers of $H^{p}$ spaces*, Proc. Amer. Math. Soc.**22**(1969), 24–27. MR**241651**, DOI https://doi.org/10.1090/S0002-9939-1969-0241651-X
---, - P. L. Duren, B. W. Romberg, and A. L. Shields,
*Linear functionals on $H^{p}$ spaces with $0<p<1$*, J. Reine Angew. Math.**238**(1969), 32–60. MR**259579** - Theodore W. Gamelin,
*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR**0410387** - T. Gamelin and G. Lumer,
*Theory of abstract Hardy spaces and the universal Hardy class*, Advances in Math.**2**(1968), 118–174. MR**226392**, DOI https://doi.org/10.1016/0001-8708%2868%2990019-4 - I. I. Privalov,
*Graničnye svoĭstva analitičeskih funkciĭ*, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR**0047765**
N. Yanagihara,

*Theory of ${H^p}$ spaces*, Pure and Appl. Math., vol. 38, Academic Press, New York, 1970. MR

**42**#3552.

*Mean growth and Taylor coefficients of some classes of functions*(to appear).

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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