Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Littlewood-Paley inequality for arbitrary rectangles in $ \mathbb{R}^2$ for  $ 0 < p \le 2$


Author: N. N. Osipov
Translated by: The author
Original publication: Algebra i Analiz, tom 22 (2010), nomer 2.
Journal: St. Petersburg Math. J. 22 (2011), 293-306
MSC (2010): Primary 42B25, 42B15
DOI: https://doi.org/10.1090/S1061-0022-2011-01141-0
Published electronically: February 8, 2011
MathSciNet review: 2668127
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The one-sided Littlewood-Paley inequality for pairwise disjoint rectangles in $ \mathbb{R}^2$ is proved for the $ L^p$-metric, $ 0 < p \le 2$. This result can be treated as an extension of Kislyakov and Parilov's result (they considered the one-dimensional situation) or as an extension of Journé's result (he considered disjoint parallelepipeds in $ \mathbb{R}^n$ but his approach is only suitable for $ p\in(1,2]$). We combine Kislyakov and Parilov's methods with methods ``dual'' to Journé's arguments.


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

  • 1. J. L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Rev. Mat. Iberoamericana 1 (1985), no. 2, 1-14. MR 0850681 (87j:42057)
  • 2. Jean-Lin Journé, Calderón-Zygmund operators on product spaces, Rev. Mat. Iberoamericana 1 (1985), no. 3, 55-91. MR 0836284 (88d:42028)
  • 3. Fernando Soria, A note on a Littlewood-Paley inequality for arbitrary intervals in $ \mathbb{R}^2$, J. London Math. Soc. (2) 36 (1987), no. 1, 137-142. MR 0897682 (88g:42026)
  • 4. J. Bourgain, On square functions on the trigonometric system, Bull. Soc. Math. Belg. Sér. B 37 (1985), no. 1, 20-26. MR 0847119 (87m:42008)
  • 5. S. V. Kislyakov and D. V. Parilov, On the Littlewood-Paley theorem for arbitrary intervals, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 327 (2005), 98-114; English transl., J. Math. Sci. (N.Y.) 139 (2006), no. 2, 6417-6424. MR 2184431 (2006h:42039)
  • 6. S. V. Kislyakov, Littlewood-Paley theorem for arbitrary intervals: weighted estimates, Zap. Nauchn. Sem. S.-Peterburg, Otdel. Mat. Inst. Steklov. (POMI) 355 (2008), 180-198; English transl. in J. Math. Sci. (N.Y.)
  • 7. Robert Fefferman, Calderón-Zygmund theory for product domains: $ H^p$ spaces, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 4, 840-843. MR 0828217 (87h:42032)
  • 8. Anthony Carbery and Andreas Seeger, $ H^p$- and $ L^p$-variants of multiparameter Calderón-Zygmund theory, Trans. Amer. Math. Soc. 334 (1992), no. 2, 719-747. MR 1072104 (93b:42035)
  • 9. R. F. Gundy and E. M. Stein, $ H^p$ theory for the poly-disc, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 3, 1026-1029. MR 0524328 (80j:32012)
  • 10. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser., vol. 43. Monogr. in Harmonic Analysis, III, Princeton Univ. Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
  • 11. S.-Y. A. Chang and R. Fefferman, The Calderón-Zygmund decomposition on product domains, Amer. J. Math. 104 (1982), no. 3, 455-468. MR 0658542 (84a:42028)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 42B25, 42B15

Retrieve articles in all journals with MSC (2010): 42B25, 42B15


Additional Information

N. N. Osipov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, 27 Fontanka, St. Petersburg 191023, Russia
Email: nicknick@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-2011-01141-0
Keywords: Littlewood–Paley inequality, Hardy class, atomic decomposition, Journé lemma, Calderón–Zygmund operator
Received by editor(s): September 11, 2009
Published electronically: February 8, 2011
Additional Notes: The author was supported by RFBR (grant no. 08-01-00358)
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society