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

 

A short proof of an inequality of Littlewood and Paley


Author: Miroslav Pavlovic
Journal: Proc. Amer. Math. Soc. 134 (2006), 3625-3627
MSC (2000): Primary 46E15
Published electronically: June 15, 2006
MathSciNet review: 2240675
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Abstract | References | Similar Articles | Additional Information

Abstract: A very short proof is given of the inequality

$\displaystyle \int_{\vert z\vert<1}\vert\nabla u(z)\vert^p (1-\vert z\vert)^{p-... ...rac 1{2\pi}\int_0^{2\pi} \vert f(e^{it})\vert^p\,dt -\vert u(0)\vert^p\right), $

where $ p>2,$ and $ u$ is the Poisson integral of $ f\in L^p(\partial \mathbb{D}),$ $ \mathbb{D}=\{z: \vert z\vert<1\}.$


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

  • 1. P.L. Duren, Theory of $ H^p$-spaces, Academic Press, New York and London, 1970. MR 0268655 (42:3552)
  • 2. J.E. Littlewood and R.E.A.C. Paley, Theorems on Fourier series and power series. II, Proc. Lond. Math. Soc. 42(1936), 52-89.
  • 3. D.H. Luecking, A new proof of an inequality of Littlewood and Paley, Proc. Amer. Math. Soc. 103(1988), 887-893. MR 0947675 (89g:30067)
  • 4. M. Pavlovic, A Littlewood-Paley theorem for subharmonic functions with subharmonic Laplacian, Publ. Inst. Math. (Belgrade) 68(82)(2000), 77-82. MR 1826098 (2002b:30040)
  • 5. P. Stein, On a theorem of M. Riesz, J. London Math. Soc. 8(1933), 52-89.

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Additional Information

Miroslav Pavlovic
Affiliation: Matematički Fakultet, Univerzitet u Beogradu, Studentski trg 16, 11000 Belgrade, Serbia, Yugoslavia
Email: pavlovic@matf.bg.ac.yu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08434-6
PII: S 0002-9939(06)08434-6
Received by editor(s): May 18, 2005
Received by editor(s) in revised form: July 11, 2005
Published electronically: June 15, 2006
Additional Notes: The author was supported by MNZŽS Grant, No. ON144010, Serbia
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.