Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A characterization of Hardy-Orlicz spaces on planar domains


Author: Manfred Stoll
Journal: Proc. Amer. Math. Soc. 117 (1993), 1031-1038
MSC: Primary 46E10; Secondary 30D55, 46J15
DOI: https://doi.org/10.1090/S0002-9939-1993-1124151-8
MathSciNet review: 1124151
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In the paper we prove that for a wide class of bounded domains $ D$ in $ \mathbb{C}$, a holomorphic function $ f$ is in the Hardy-Orlicz space $ {H_\phi }(D)$ if and only if

$\displaystyle \iint_D {\delta (z)\phi ''(\log \vert f(z)\vert)\frac{{\vert f'(z){\vert^2}}} {{\vert f(z){\vert^2}}}dx\,dy < \infty ,}$

where $ \delta (z)$ denotes the distance from $ z$ to the boundary of $ D$ and $ \phi $ is a strongly convex function on $ ( - \infty ,\infty )$ for which $ \phi ''(t)$ exists for all $ t$.

References [Enhancements On Off] (What's this?)

  • [F] S. D. Fisher, Function theory on planar domains, John Wiley & Sons, New York, 1983. MR 694693 (85d:30001)
  • [KS] S. Kobayashi and N. Suita, Area integrals and $ {H_p}$ norms of analytic functions, Complex Variables 5 (1986), 181-188. MR 846486 (87i:30070)
  • [R1] W. Rudin, Analytic functions of class $ {H_p}$, Trans. Amer. Math. Soc. 78 (1955), 46-66. MR 0067993 (16:810b)
  • [R2] -, Function theory in polydiscs, W. A. Benjamin, Inc, New York, 1969. MR 0255841 (41:501)
  • [T] M. Tsuji, Potential theory, Chelsea Publ. Co., New York, 1975. MR 0414898 (54:2990)
  • [W] E. Wojcicka, Functions of bounded characteristic in multiply connected domains, Doctoral Dissertation, Univ. of South Carolina, 1985.
  • [Y1] S. Yamashita, Criteria for functions to be of Hardy class $ {H^p}$, Proc. Amer. Math. Soc. 75 (1979), 69-72. MR 529215 (80h:30031)
  • [Y2] -, Holomorphic functions and area integrals, Boll. Un. Math. Ital. A (1) 6 (1982), 115-120. MR 652371 (83e:30038)
  • [Z] A. Zygmund, Trigonometric series, Cambridge Univ. Press, London, 1968. MR 0236587 (38:4882)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E10, 30D55, 46J15

Retrieve articles in all journals with MSC: 46E10, 30D55, 46J15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1124151-8
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society