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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Estimates on the mean growth of $H^p$ functions in convex domains of finite type


Author: Hong Rae Cho
Journal: Proc. Amer. Math. Soc. 131 (2003), 2393-2398
MSC (2000): Primary 32A35, 32A26; Secondary 32T25
Published electronically: March 17, 2003
MathSciNet review: 1974636
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Abstract: Let $D$ be a bounded convex domain of finite type in $\mathbb C^n$with smooth boundary. In this paper, we prove the following inequality:

\begin{displaymath}\left(\int_0^{\delta_0}\mathcal M_q^\lambda(f;t)~ t^{\lambda ... ...p-1/q)-1}dt\right)^{1/\lambda}\leq C_{p,q}\Vert f\Vert _{p,0}, \end{displaymath}

where $1<p<q<\infty, f\in H^p(D)$, and $p\leq\lambda<\infty$. This is a generalization of some classical result of Hardy-Littlewood for the case of the unit disc. Using this inequality, we can embed the $H^p$ space into a weighted Bergman space in a convex domain of finite type.


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Additional Information

Hong Rae Cho
Affiliation: Department of Mathematics Education, Andong National University, Andong 760-749, South Korea
Address at time of publication: Department of Mathematics, Pusan National University, Pusan 609-735, Korea
Email: chohr@anu.ac.kr, chohr@pusan.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07012-6
PII: S 0002-9939(03)07012-6
Keywords: Mean growth of $H^p$ functions, convex domains of finite type, reproducing kernel
Received by editor(s): March 4, 2002
Published electronically: March 17, 2003
Additional Notes: The author was supported by grant No. R01-2000-000-00001-0 from the Basic Research Program of the Korea Science & Engineering Foundation. The author thanks the referee for helpful suggestions.
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2003 American Mathematical Society