# Norm and Taylor coefficients estimates of holomorphic functions in balls

• Volume: 54, Issue: 3, page 271-297
• ISSN: 0066-2216

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## Abstract

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A classical result of Hardy and Littlewood states that if $f\left(z\right)={\sum }_{m=0}^{\infty }{a}_{m}{z}^{m}$ is in ${H}^{p}$, 0 < p ≤ 2, of the unit disk of ℂ, then ${\sum }_{m=0}^{\infty }{\left(m+1\right)}^{p-2}{|{a}_{m}|}^{p}\le {c}_{p}\parallel f{\parallel }_{p}^{p}$ where ${c}_{p}$ is a positive constant depending only on p. In this paper, we provide an extension of this result to Hardy and weighted Bergman spaces in the unit ball of ${ℂ}^{n}$, and use this extension to study some related multiplier problems in ${ℂ}^{n}$.

## How to cite

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Jacob Burbeam, and Do Young Kwak. "Norm and Taylor coefficients estimates of holomorphic functions in balls." Annales Polonici Mathematici 54.3 (1991): 271-297. <http://eudml.org/doc/262478>.

@article{JacobBurbeam1991,
abstract = {A classical result of Hardy and Littlewood states that if $f(z) = ∑_\{m=0\}^\{∞\} a_m z^m$ is in $H^p$, 0 < p ≤ 2, of the unit disk of ℂ, then $∑_\{m=0\}^\{∞\} (m+1)^\{p-2\}|a_m|^p ≤ c_p ∥f∥_p^p$ where $c_p$ is a positive constant depending only on p. In this paper, we provide an extension of this result to Hardy and weighted Bergman spaces in the unit ball of $ℂ^n$, and use this extension to study some related multiplier problems in $ℂ^n$.},
author = {Jacob Burbeam, Do Young Kwak},
journal = {Annales Polonici Mathematici},
keywords = {holomorphic functions; unit ball; growth estimates; Taylor coefficients},
language = {eng},
number = {3},
pages = {271-297},
title = {Norm and Taylor coefficients estimates of holomorphic functions in balls},
url = {http://eudml.org/doc/262478},
volume = {54},
year = {1991},
}

TY - JOUR
AU - Jacob Burbeam
AU - Do Young Kwak
TI - Norm and Taylor coefficients estimates of holomorphic functions in balls
JO - Annales Polonici Mathematici
PY - 1991
VL - 54
IS - 3
SP - 271
EP - 297
AB - A classical result of Hardy and Littlewood states that if $f(z) = ∑_{m=0}^{∞} a_m z^m$ is in $H^p$, 0 < p ≤ 2, of the unit disk of ℂ, then $∑_{m=0}^{∞} (m+1)^{p-2}|a_m|^p ≤ c_p ∥f∥_p^p$ where $c_p$ is a positive constant depending only on p. In this paper, we provide an extension of this result to Hardy and weighted Bergman spaces in the unit ball of $ℂ^n$, and use this extension to study some related multiplier problems in $ℂ^n$.
LA - eng
KW - holomorphic functions; unit ball; growth estimates; Taylor coefficients
UR - http://eudml.org/doc/262478
ER -

## References

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1. [1] F. Beatrous and J. Burbea, Holomorphic Sobolev spaces on the ball, Dissertationes Math. 276 (1989). Zbl0691.46024
2. [2] R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. Zbl0326.32011
3. [3] P. L. Duren, Theory of ${H}^{p}$ Spaces, Academic Press, New York 1970. Zbl0215.20203
4. [4] P. L. Duren and A. L. Shields, Properties of ${H}^{p}$ (0 < p < 1) and its containing Banach space, Trans. Amer. Math. Soc. 141 (1969), 255. Zbl0181.40401
5. [5] P. L. Duren and A. L. Shields,Coefficient multipliers of ${H}^{p}$ and ${B}^{p}$ spaces, Pacific J. Math. 32 (1970), 69-78.
6. [6] T. M. Flett, On the rate of growth of mean values of holomorphic and harmonic functions, Proc. London Math. Soc. 20 (1970), 749.
7. [7] F. Forelli and W. Rudin, Projections on the spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974), 593-602. Zbl0297.47041
8. [8] G. H. Hardy and J. E. Littlewood, Theorems concerning mean values of analytic or harmonic functions, Quart. J. Math. Oxford Ser. 12 (1942), 221-256. Zbl0060.21702
9. [9] A. Korányi and S. Vagi, Singular integrals in homogeneous spaces and some problems of classical analysis, Ann. Scuola Norm. Sup. Pisa 25 (1971), 575-648. Zbl0291.43014
10. [10] E. M. Stein and G. Weiss, On the interpolation of analytic families of operators acting on ${H}^{p}$ spaces, Tôhoku Math. J. (2) 9 (1957), 318-339. Zbl0083.34201

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