On the Generalized Marcinkiewicz Integral Operators with Rough Kernels

Fan, Dashan ; Wu, Huoxiong

Canadian mathematical bulletin, Tome 54 (2011) no. 1, pp. 100-112

Résumé

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.

Détail

DOI : 10.4153/CMB-2010-085-3

Mots-clés : 42B20, 42B25, 42B30, 42B99, Marcinkiewicz integral, Littlewood–Paley theory, Triebel–Lizorkin space, rough kernel, product domain

Bibliographie

[1] [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/sm167-3-4 Google Scholar

[2] [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] [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-O Google Scholar

[4] [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/sm153-1-4 Google Scholar

[5] [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)00419-5 Google Scholar

[6] [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.2001.7465 Google Scholar

[7] [7] Ding, Y., L 2 -boundedness of Marcinkiewicz integral with rough kernel. Hokkaido Math. J. 27(1998), no. 1, 105–115. Google Scholar

[8] [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/1022636747 Google Scholar

[9] [9] Duoandikoetxea, J., Multiple singular integrals and maximal functions along hypersurfaces. Ann. Inst. Fourier (Grenoble) 36(1986), no. 4, 185–206. Google Scholar

[10] [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/1178207481 Google Scholar

[11] [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] [12] Stein, E. M., On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans. Amer. Math. Soc. 88(1958), 430–466. doi:10.2307/1993226 Google Scholar

[13] [13] Triebel, H., Theory of function spaces. Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. Google Scholar

[14] [14] Walsh, T., On the function of Marcinkiewicz. Studia Math. 44(1972), 203–217. Google Scholar

[15] [15] Wang, M., Some Problems on singular integrals on product spaces. Ph. D. Thesis (in Chinese), Zhejiang Univ., Hangzhou, 2002. Google Scholar

[16] [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.200310369 Google Scholar

[17] [17] Wu, H., A rough multiple Marcinkiewicz integral along continuous surfaces. Tohoku Math. J. 59(2007), no. 2, 145–166. doi:10.2748/tmj/1182180732 Google Scholar

[18] [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-z Google Scholar

[19] [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.2005.08.087 Google Scholar