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On the Generalized Marcinkiewicz Integral Operators with Rough Kernels

Published online by Cambridge University Press:  20 November 2018

Dashan Fan  and
Huoxiong Wu
Dashan Fan
Affiliation:
Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI, U.S.A.e-mail: fan@csd.uwm.edu
Huoxiong Wu
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen Fujian, P. R. Chinae-mail: huoxwu@xmu.edu.cn
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Abstract

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A class of generalized Marcinkiewicz integral operators is introduced, and, under rather weak conditions on the integral kernels, the boundedness of such operators on ${{L}^{p}}$ and Triebel–Lizorkin spaces is established.

Keywords

42B2042B2542B3042B99Marcinkiewicz integralLittlewood–Paley theoryTriebel–Lizorkin spacerough kernelproduct domain

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 1 , 01 March 2011 , pp. 100 - 112
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Al-Qassem, H., Al-Salman, A., Cheng, L. C., and Pan, Y., Marcinkiewicz integrals on product spaces. Studia Math. 167(2005), no. 3, 227–234. doi:10.4064/sm167-3-4Google Scholar
[2] Al-Qassem, H., Al-Salman, A., 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
[3] Chen, J., Ding, Y., and Fan, D., Certain square functions on product spaces. Math. Nachr. 230(2001), 5–18. doi:10.1002/1522-2616(200110)230:1h5::AID-MANA5i3.0.CO;2-OGoogle Scholar
[4] Chen, J., Fan, D., and Ying, Y., The method of rotation and Marcinkiewicz integrals on product domains. Studia Math. 153(2002), no. 1, 41–58. doi:10.4064/sm153-1-4Google Scholar
[5] Chen, J., Fan, D., and Ying, Y., Singular integral operators on function spaces. J. Math. Anal. Appl. 276(2002), no. 2, 691–708. doi:10.1016/S0022-247X(02)00419-5Google Scholar
[6] Choi, Y., Marcinkiewicz integrals with rough homogeneous kernels of degree zero in product domains. J. Math. Anal. Appl. 261(2001), no. 1, 53–60. doi:10.1006/jmaa.2001.7465Google Scholar
[7] Ding, Y., L 2 -boundedness of Marcinkiewicz integral with rough kernel. Hokkaido Math. J. 27(1998), no. 1, 105–115.Google Scholar
[8] Ding, Y., Fan, D., and Pan, Y., On the Lp-boundedness of Marcinkiewicz integrals. Michigan Math. J. 50(2002), no. 1, 17–26. doi:10.1307/mmj/1022636747Google Scholar
[9] Duoandikoetxea, J., Multiple singular integrals and maximal functions along hypersurfaces. Ann. Inst. Fourier (Grenoble) 36(1986), no. 4, 185–206.Google Scholar
[10] 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. doi:10.2748/tmj/1178207481Google Scholar
[11] Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.Google Scholar
[12] Stein, E. M., On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans. Amer. Math. Soc. 88(1958), 430–466. doi:10.2307/1993226Google Scholar
[13] Triebel, H., Theory of function spaces. Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983.Google Scholar
[14] Walsh, T., On the function of Marcinkiewicz. Studia Math. 44(1972), 203–217.Google Scholar
[15] Wang, M., Some Problems on singular integrals on product spaces. Ph. D. Thesis (in Chinese), Zhejiang Univ., Hangzhou, 2002.Google Scholar
[16] Wu, H., General Littlewood-Paley functions and singular integral operators on product spaces. Math. Nachr. 279(2006), no. 4, 431–444. doi:10.1002/mana.200310369Google Scholar
[17] Wu, H., A rough multiple Marcinkiewicz integral along continuous surfaces. Tohoku Math. J. 59(2007), no. 2, 145–166. doi:10.2748/tmj/1182180732Google Scholar
[18] Wu, H., On Marcinkiewicz integral operators with rough kernels. Integral Equations Operator Theory 52(2005), no. 2, 285–298. doi:10.1007/s00020-004-1339-zGoogle Scholar
[19] Wu, H., Lp bounds for Marcinkiewicz integrals associated to surfaces of revolution. J. Math. Anal. Appl. 321(2006), no. 2, 811–827. doi:10.1016/j.jmaa.2005.08.087Google Scholar