Littlewood–Paley inequality for arbitrary rectangles in $\mathbb {R}^2$ for $0 < p \le 2$
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N. N. Osipov
Translated by: The author - St. Petersburg Math. J. 22 (2011), 293-306
- DOI: https://doi.org/10.1090/S1061-0022-2011-01141-0
- Published electronically: February 8, 2011
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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
- José L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Rev. Mat. Iberoamericana 1 (1985), no. 2, 1–14. MR 850681, DOI 10.4171/RMI/7
- Jean-Lin Journé, Calderón-Zygmund operators on product spaces, Rev. Mat. Iberoamericana 1 (1985), no. 3, 55–91. MR 836284, DOI 10.4171/RMI/15
- Fernando Soria, A note on a Littlewood-Paley inequality for arbitrary intervals in $\textbf {R}^2$, J. London Math. Soc. (2) 36 (1987), no. 1, 137–142. MR 897682, DOI 10.1112/jlms/s2-36.1.137
- J. Bourgain, On square functions on the trigonometric system, Bull. Soc. Math. Belg. Sér. B 37 (1985), no. 1, 20–26. MR 847119
- 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), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 33, 98–114, 236–237 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 139 (2006), no. 2, 6417–6424. MR 2184431, DOI 10.1007/s10958-006-0359-4
- 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.)
- 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 828217, DOI 10.1073/pnas.83.4.840
- 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, DOI 10.1090/S0002-9947-1992-1072104-4
- 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 524328, DOI 10.1073/pnas.76.3.1026
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Sun-Yung A. Chang and Robert Fefferman, The Calderón-Zygmund decomposition on product domains, Amer. J. Math. 104 (1982), no. 3, 455–468. MR 658542, DOI 10.2307/2374150
Bibliographic Information
- N. N. Osipov
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, 27 Fontanka, St. Petersburg 191023, Russia
- Email: nicknick@pdmi.ras.ru
- 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)
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2668127