Geodesic
Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces
Canadian mathematical bulletin, Tome 55 (2012) no. 3, pp. 646-662

Voir la notice de l'article provenant de la source Cambridge University Press

Under the assumption that $\mu $ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a Hörmander-type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case.
DOI : 10.4153/CMB-2011-139-1
Mots-clés : 42B25, 47B47, 42B20, 47A30, non doubling measure, Marcinkiewicz integral, commutator, Lipβ (μ), H 1(μ) 
Zhou, Jiang; Ma, Bolin. Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces. Canadian mathematical bulletin, Tome 55 (2012) no. 3, pp. 646-662. doi: 10.4153/CMB-2011-139-1
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-139-1/}
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[1] [1] Al-Salman, A., Al-Qassem, H., Cheng, L. C., and Pan, Y., Lp bounds for the function of Marcinkiewicz. Math. Res. Lett. 9(2002), no. 5-6, 697–700. Google Scholar

[2] [2] Ding, Y., Lu, S., and Xue, Q., Marcinkiewicz integral on Hardy spaces. Integral Equations Operator Theory, 42(2002), no. 2, 174–182. Google Scholar | DOI

[3] [3] Ding, Y., Lu, S., and Zhang, P., Weighted weak type estimates for commutators of the Marcinkiewicz integrals. Sci. China Ser. A., 47(2004), no. 1, 83–95. Google Scholar | DOI

[4] [4] Fan, D. and Sato, S.,Weak type (1, 1) estimates for Marcinkiewicz integrals with rough kernels. Tohoku Math. J. 53(2001), no. 2, 265–284. Google Scholar | DOI

[5] [5] García-Cuerva, J. and Gatto, A. E., Boundedness properties of fractional integral operators associated to non-doubling measures. Studia Math. 162(2004), no. 3, 245–261 Google Scholar | DOI

[6] [6] García-Cuerva, J. and Gatto, A. E., Lipschitz spaces and Calderón-Zygmund operators associated to non-doubling measures. Publ. Mat. 49(2005), no. 2, 285–296. Google Scholar

[7] [7] Hu, G., Meng, Y., and Yang, D., Multilinear commutators of singular integrals with non doubling measure. Integral Equations Operator Theory 51(2005), no. 2, 235–255. Google Scholar | DOI

[8] [8] Hu, G. and Yan, D., On the commutator of the Marcinkiewicz integral. J. Math. Anal. Appl. 283(2003), no. 2, 351–361 (2003) Google Scholar | DOI

[9] [9] Hu, G., Lin, H., and Yang, D., Marcinkiewicz integrals with non-doubling measures. Integral Equations Operator Theory, 58(2007), no. 2, 205–238, Google Scholar | DOI

[10] [10] Li, L. and Jiang, Y.-S., Estimates for maximal multilinear commutators on non-homogeneous spaces. J. Math. Anal. Appl. 355(2009), no. 1, 243–257. Google Scholar | DOI

[11] [11] Lorente, M., Riveros, M. S., and de la Torre, A., Weighted estimates for singular integral operators satisfying Hörmander's conditions of Young type. J. Fourier Anal. Appl. 11(2005), no. 5, 497–509. Google Scholar | DOI

[12] [12] Lu, S., Ding, Y., and Yan, D., Singular Integral and Related Topics. World Scientific Publishing Company, Hackensak, NJ, 2007. Google Scholar

[13] [13] Marcinkiewicz, J., Sur quelques intégrales du type de Dini. Ann. Soc. Polon. Math. 17(1938), 42–50. Google Scholar

[14] [14] Meng, Y. and Yang, D., Boundedness of commutators with Lipschitz functions in non-homogeneous spaces. Taiwanese J Math. 10(2006), no. 6, 1443–1464. Google Scholar

[15] [15] Mo, H. and Lu, S., Boundedness of generalized higher commutators of Marcinkiewicz integrals. Acta Math. Sci. Ser. B Engl. Ed. 27(2007) no. 4, 852–866. Google Scholar

[16] [16] Sakamoto, N. and Yabuta, K., Boundedness of Marcinkiewicz functions. Studia. Math. 135(1999), no. 2, 103–142. Google Scholar

[17] [17] Stein, E. M., On the functions of Littlewood–Paley, Lusin, and Marcinkiewicz. Trans. Amer. Math. Soc. 88(1958), 430–466. Google Scholar | DOI

[18] [18] Torchinsky, A. and Wang, S., A note on the Marcinkiewicz integral. Colloq. Math., 60/61(1990), no. 1, 235–243. Google Scholar

[19] [19] Tolsa, X., BMO, H 1 and Calderón-Zygmund operators for non doubling measures. Math. Ann. 319(2001), no. 1, 89–149. Google Scholar | DOI

[20] [20] Tolsa, X., The space H 1 for nondoubling measures in terms of a grand maximal operator. Trans. Amer. Math. Soc. 355(2003), no. 1, 315–348. Google Scholar | DOI

[21] [21] Tolsa, X., Littlewood-Paley theory and the T(1) theorem with non-doubling measures. Adv. Math. 164(2001), no. 1, 57–116. Google Scholar | DOI

[22] [22] Tolsa, X., Painlevé's problem and the semiadditivity of analytic capacity. Acta Math. 190(2003), no. 1, 105–149. Google Scholar | DOI

[23] [23] Wu, H., On Marcinkiewicz integral operators with rough kernels. Integral Equations Operator Theory 52(2005), no. 2, 285–298. Google Scholar | DOI

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