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MULTIPLIERS AND CHARACTERIZATION OF THE DUAL OF NEVANLINNA-TYPE SPACES

Part of: Linear function spaces and their duals Spaces and algebras of analytic functions Analytic spaces

Published online by Cambridge University Press:  07 September 2023

MIECZYSŁAW MASTYŁO Open the ORCID record for MIECZYSŁAW MASTYŁO [Opens in a new window]  and
BARTOSZ STANIÓW Open the ORCID record for BARTOSZ STANIÓW [Opens in a new window]
MIECZYSŁAW MASTYŁO*
Affiliation:
Faculty of Mathematics and Computer Science Adam Mickiewicz University Uniwersytetu Poznańskiego 4 61-614 Poznań Poland
BARTOSZ STANIÓW
Affiliation:
Faculty of Mathematics and Computer Science Adam Mickiewicz University Uniwersytetu Poznańskiego 4 61-614 Poznań Poland bs21276@amu.edu.pl
*
e-mail: mieczyslaw.mastylo@amu.edu.pl
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Abstract

The Nevanlinna-type spaces $N_\varphi $ of analytic functions on the disk in the complex plane generated by strongly convex functions $\varphi $ in the sense of Rudin are studied. We show for some special class of strongly convex functions asymptotic bounds on the growth of the Taylor coefficients of a function in $N_\varphi $ and use these to characterize the coefficient multipliers from $N_\varphi $ into the Hardy spaces $H^p$ with $0<p\leqslant \infty $. As a by-product, we prove a representation of continuous linear functionals on $N_\varphi $.

MSC classification

Primary: 46E10: Topological linear spaces of continuous, differentiable or analytic functions 32C37: Duality theorems 30H15: Nevanlinna class and Smirnov class

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Type
Article
Information
Nagoya Mathematical Journal , Volume 253 , March 2024 , pp. 216 - 240
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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Footnotes

M.M. was supported by the National Science Center, Poland, project no. 2019/33/B/ST1/00165.

References

Duren, P., Theory of ${H}^p$ Spaces, Pure and Applied Mathematics, 38, Academic Press, New York, 1970.Google Scholar
Duren, P., Romberg, P., and Shields, B. W., Linear functionals on ${H}^p$ spaces with $0<p<1$ , J. Reine Angew. Math. 238 (1969), 32–60.Google Scholar
Dyakonov, K. M., Smooth functions in the range of a Hankel operator , Indiana Univ. Math. J. 43 (1994), no. 3, 805–838.CrossRefGoogle Scholar
Garnett, J. B., Bounded Analytic Functions, Academic Press, New York, 1981.Google Scholar
Hasumi, M. and Kataoka, S., Remarks on Hardy–Orlicz classes , Arch. Math. 51 (1988), 455–463.CrossRefGoogle Scholar
Kalton, N. J., Peck, N. T., and Roberts, J. W., An $F$ -Space Sampler, London Mathematical Society Lecture Notes Series, 89, Cambridge University Press, Cambridge, 1984.Google Scholar
Krein, S. G., Petunin, Yu. I., and Semenov, E. M., Interpolation of Linear Operators, Nauka, Moscow, 1978; English transl. in American Mathematical Society, Providence, 1982.Google Scholar
Leśniewicz, R., On linear functionals in Hardy–Orlicz spaces, $\mathrm{I}$ , Studia Math. 46 (1973), 53–77.CrossRefGoogle Scholar
Matuszewska, W. and Orlicz, W., On certain properties of $\varphi$ -functions , Bull. Acad. Polon. Sci., Sèr. Sci. Math. Astronom. Phys. 8 (1960), 439–443.Google Scholar
Matuszewska, W. and Orlicz, W., On some classes of functions with regard to their orders of growth , Studia Math. 26 (1965), 11–24.CrossRefGoogle Scholar
McCarthy, J. E., Topologies on the Smirnov class , J. Funct. Anal. 104 (1992), no. 1, 229–241.CrossRefGoogle Scholar
Meshtrovich, R. and Subbotin, A. V., Multipliers and linear functionals for I. I. Privalov spaces of holomorphic functions in the unit disc , Dokl. Akad. Nauk 365 (1999), no. 4, 452–454 (in Russian).Google Scholar
Nawrocki, M., On the solid hulls of the Nevanlinna and the Smirnov classes , Ann. Acad. Sci. Fenn. Math. 39 (2014), no. 2, 897–904.CrossRefGoogle Scholar
Privalov, I. I., Boundary Properties of Analytic Functions, Gostekhizdat, Moscow, 1941.Google Scholar
Rosenblum, M. and Rovnyak, J., Hardy Classes and Operator Theory, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1985.Google Scholar
Rudin, W., Function Theory in Polydiscs, W. A. Benjamin Inc., New York, 1970.Google Scholar
Shapiro, J. H. and Shields, A. L., Unusual topological properties of the Nevanlinna class , Amer. J. Math. 97 (1976), 915–936.CrossRefGoogle Scholar
Smirnow, V. I., Sur les valeurs limits des fonctions, reguli’ere ’a l’int’erieur d’un cercle , J. Soc. Phys. Math. L’eningrade 2 (1929), 22–37.Google Scholar
Stoll, M., Harmonic majorants for plurisubharmonic functions on bounded symmetric domains with application to the spaces ${H}_F$ and ${N}_{\ast }$ , J. Reine Angew. Math. 282 (1976), 80–87.Google Scholar
Stoll, M., Mean growth and Taylor coefficients of some topological algebras of analytic functions , Ann. Pol. Math. 35 (1977), 139–158.CrossRefGoogle Scholar
Yanagihara, N., Multipliers and linear functionals for the class ${N}^{+}$ , Trans. Amer. Math. Soc. 180 (1973), 449–461.Google Scholar
Yanagihara, N., Mean growth and Taylor coefficients of some classes of functions , Ann. Pol. Math. 30 (1974), 37–48.CrossRefGoogle Scholar
Zayed, A. I., Topological vector spaces of analytic functions , Complex Var. Theory Appl. 2 (1983), 27–50.Google Scholar