Skip to main content Accessibility help
×
Home

Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces

  • Jiang Zhou (a1) and Bolin Ma (a2)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces
      Available formats
      ×

Copyright

References

Hide All
[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, 697700.
[2] Ding, Y., Lu, S., and Xue, Q., Marcinkiewicz integral on Hardy spaces. Integral Equations Operator Theory, 42(2002), no. 2, 174182. http://dx.doi.org/10.1007/BF01275514
[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, 8395. http://dx.doi.org/10.1360/03ys0084
[4] Fan, D. and Sato, S.,Weak type (1, 1) estimates for Marcinkiewicz integrals with rough kernels. Tohoku Math. J. 53(2001), no. 2, 265284. http://dx.doi.org/10.2748/tmj/1178207481
[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, 245261 http://dx.doi.org/10.4064/sm162-3-5
[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, 285296.
[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, 235255. http://dx.doi.org/10.1007/s00020-003-1251-y
[8] Hu, G. and Yan, D., On the commutator of the Marcinkiewicz integral. J. Math. Anal. Appl. 283(2003), no. 2, 351361 (2003) http://dx.doi.org/10.1016/S0022-247X(02)00498-5
[9] Hu, G., Lin, H., and Yang, D., Marcinkiewicz integrals with non-doubling measures. Integral Equations Operator Theory, 58(2007), no. 2, 205238, http://dx.doi.org/10.1007/s00020-007-1481-5
[10] Li, L. and Jiang, Y.-S., Estimates for maximal multilinear commutators on non-homogeneous spaces. J. Math. Anal. Appl. 355(2009), no. 1, 243257. http://dx.doi.org/10.1016/j.jmaa.2009.01.022
[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, 497509. http://dx.doi.org/10.1007/s00041-005-4039-4
[12] Lu, S., Ding, Y., and Yan, D., Singular Integral and Related Topics. World Scientific Publishing Company, Hackensak, NJ, 2007.
[13] Marcinkiewicz, J., Sur quelques intégrales du type de Dini. Ann. Soc. Polon. Math. 17(1938), 4250.
[14] Meng, Y. and Yang, D., Boundedness of commutators with Lipschitz functions in non-homogeneous spaces. Taiwanese J Math. 10(2006), no. 6, 14431464.
[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, 852866.
[16] Sakamoto, N. and Yabuta, K., Boundedness of Marcinkiewicz functions. Studia. Math. 135(1999), no. 2, 103142.
[17] Stein, E. M., On the functions of Littlewood–Paley, Lusin, and Marcinkiewicz. Trans. Amer. Math. Soc. 88(1958), 430466. http://dx.doi.org/10.1090/S0002-9947-1958-0112932-2
[18] Torchinsky, A. and Wang, S., A note on the Marcinkiewicz integral. Colloq. Math., 60/61(1990), no. 1, 235243.
[19] Tolsa, X., BMO, H 1 and Calderón-Zygmund operators for non doubling measures. Math. Ann. 319(2001), no. 1, 89149. http://dx.doi.org/10.1007/PL00004432
[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, 315348. http://dx.doi.org/10.1090/S0002-9947-02-03131-8
[21] Tolsa, X., Littlewood-Paley theory and the T(1) theorem with non-doubling measures. Adv. Math. 164(2001), no. 1, 57116. http://dx.doi.org/10.1006/aima.2001.2011
[22] Tolsa, X., Painlevé's problem and the semiadditivity of analytic capacity. Acta Math. 190(2003), no. 1, 105149. http://dx.doi.org/10.1007/BF02393237
[23] Wu, H., On Marcinkiewicz integral operators with rough kernels. Integral Equations Operator Theory 52(2005), no. 2, 285298. http://dx.doi.org/10.1007/s00020-004-1339-z
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces

  • Jiang Zhou (a1) and Bolin Ma (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed