Geodesic
Littlewood--Paley inequality for arbitrary rectangles in $\mathbb R^2$ for $0$
Algebra i analiz, Tome 22 (2010) no. 2, pp. 164-184.

Voir la notice de l'article provenant de la source Math-Net.Ru

The one-sided Littlewood–Paley inequality for pairwise disjoint rectangles in $\mathbb R^2$ is proved for the $L^p$-metric, $0$. 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.
Keywords: Littlewood–Paley inequality, Hardy class, atomic decomposition, Journé lemma, Calderón–Zygmund operator. 
@article{AA_2010_22_2_a4,
     author = {N. N. Osipov},
     title = {Littlewood--Paley inequality for arbitrary rectangles in $\mathbb R^2$ for $0<p\le2$},
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     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2010_22_2_a4/}
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N. N. Osipov. Littlewood--Paley inequality for arbitrary rectangles in $\mathbb R^2$ for $0

[1] Rubio de Francia J. L., “A Littlewood–Paley inequality for arbitrary intervals”, Rev. Mat. Iberoamericana, 1:2 (1985), 1–14 | MR | Zbl

[2] Journé Jean-Lin, “Calderón–Zygmund operators on product spaces”, Rev. Mat. Iberoamericana, 1:3 (1985), 55–91 | MR | Zbl

[3] Soria Fernando, “A note on a Littlewood–Paley inequality for arbitrary intervals in $\mathbb R^2$”, J. London Math. Soc. (2), 36:1 (1987), 137–142 | DOI | MR | Zbl

[4] Bourgain J., “On square functions on the trigonometric system”, Bull. Soc. Math. Belg. Sér. B, 37:1 (1985), 20–26 | MR | Zbl

[5] Kislyakov S. V., Parilov D. V., “O teoreme Litlvuda–Peli dlya proizvolnykh intervalov”, Zap. nauch. semin. POMI, 327, 2005, 98–114 | MR | Zbl

[6] Kislyakov S. V., “Teorema Litlvuda–Peli dlya proizvolnykh intervalov: vesovye otsenki”, Zap. nauch. semin. POMI, 355, 2008, 180–198

[7] Fefferman Robert, “Calderón–Zygmund theory for product domains: $H^p$ spaces”, Proc. Nat. Acad. Sci. USA, 83:4 (1986), 840–843 | DOI | MR | Zbl

[8] Carbery Anthony, Seeger Andreas, “$H^p$- and $L^p$-variants of multiparameter Calderón–Zygmund theory”, Trans. Amer. Math. Soc., 334:2 (1992), 719–747 | DOI | MR | Zbl

[9] Gundy R. F., Stein E. M., “$H^p$ theory for the poly-disc”, Proc. Nat. Acad. Sci. USA, 76:3 (1979), 1026–1029 | DOI | MR | Zbl

[10] Stein Elias M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser. Monogr. in Harmonic Analysis, III, 43, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl

[11] Chang S.-Y. A., Fefferman R., “The Calderón–Zygmund decomposition on product domains”, Amer. J. Math., 104:3 (1982), 455–468 | DOI | MR | Zbl