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BOUNDEDNESS OF MULTIPLE MARCINKIEWICZ INTEGRAL OPERATORS WITH ROUGH KERNELS

  • Wu Huoxiong (School of Mathematical Sciences Xiamen University)
  • Published : 2006.05.01

Abstract

This paper is concerned with giving some rather weak size conditions implying the $L^P$ boundedness of the multiple Marcin-kiewicz integrals for some fixed $1\;<\;p\;<\;{\infty}$, which essentially improve and extend some known results.

Keywords

References

  1. A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139 https://doi.org/10.1007/BF02392130
  2. J. Chen, Y. Ding, and D. Fan, $L^p$ boundedness of the rough Marcinkiewicz integral on product domains, Chinese J. Contemp. Math. 21 (2000), no. 1, 47-54
  3. J. Chen, Y. Ding, and D. Fan, Certain square functions on product spaces, Math. Nachr. 230 (2001), 5-18 https://doi.org/10.1002/1522-2616(200110)230:1<5::AID-MANA5>3.0.CO;2-O
  4. J. Chen, D. Fan, and Y. Ying, Rough Marcinkiewicz integrals with $L(log^+L)^2$ kernels on product spaces, Adv. Math. (China) 30 (2001), no. 2, 179-181
  5. J. Chen, D. Fan, and Y. Ying, The method of rotation and Marcinkiewicz integrals on product domains, Studia Math. 153 (2002), no. 1, 41-58 https://doi.org/10.4064/sm153-1-4
  6. S. Chanillo and R. L. Wheeden, Inequalities for Peano maximal functions and Marcinkiewicz integrals, Duke Math. J. 50 (1983), no. 3, 573-603 https://doi.org/10.1215/S0012-7094-83-05027-5
  7. S. Chanillo and R. L. Wheeden, Relations between Peano derivatives and Marcinkiewicz integrals, in: Con- ference on harmonic analysis in honor of Antoni Zygmund, Vols. I, II (Chicago, Ill., 1981), 508-525, Wadsworth Math. Ser. Wadsworth, 1983
  8. Y. Choi, Marcinkiewicz integrals with rough homogeneous kernels of degree zero in product domains, J. Math. Anal. Appl. 261 (2001), no. 1, 53-60 https://doi.org/10.1006/jmaa.2001.7465
  9. Y. Ding, $L^2$-boundedness of Marcinkiewicz integral with rough kernel, Hokkaido Math. J. 27 (1998), no. 1, 105-115 https://doi.org/10.14492/hokmj/1351001253
  10. J. Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Inst. Fourier (Gronble) 36 (1986), no. 4, 185-206 https://doi.org/10.5802/aif.1073
  11. R. Fefferman and E. M. Stein, Singular integrals on product spaces, Adv. Math. 45 (1982), no. 2, 117-143 https://doi.org/10.1016/S0001-8708(82)80001-7
  12. L. Grafakos and A. Stefanov, $L^p$ bounds for singular integrals and maximal singular integrals with rough kernels, Indiana Univ. Math. J. 47 (1998), no. 2, 455-469
  13. G. Hu, S. Lu, and D. Yan, $L^p(\mathbb{R}^m\;\times\;\mathbb{R}^n)$ boundedness for the Marcinkiewicz integral on product spaces, Sci. China Ser. A 46 (2003), no. 1, 75-82 https://doi.org/10.1360/03ys9008
  14. L. Hormander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math. 104 (1960), 93-140 https://doi.org/10.1007/BF02547187
  15. M. Sakamoto and K. Yabuta, Boundedness of Marcinkiewicz functions, Studia Math. 135 (1999), no. 2, 103-142
  16. S. Sato, Remarks on square functions in the Littlewood-Paley theory, Bull. Austral. Math. Soc. 58 (1998), no. 2, 199-211 https://doi.org/10.1017/S0004972700032172
  17. E. M. Stein, On the function of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466 https://doi.org/10.2307/1993226
  18. E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integral, Princeton Univ. Press, Princeton, NJ, 1993
  19. E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proc. Internat. Congr. Math., Berkeley (1986), 196-221
  20. E. M. Stein, Some geometrical concepts arising in harmonic analysis, Geom. Funct. Anal. Special Vol. (2000), 434-453
  21. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, New Jersey, 1970
  22. T. Walsh, On the function of Marcinkiewicz, Studia Math. 44 (1972), 203-217 https://doi.org/10.4064/sm-44-3-203-217
  23. H. Wu, On Marcinkiewicz integral operators with rough kernels, Integral Equations Operator Theory 52 (2005), no. 2, 285-298 https://doi.org/10.1007/s00020-004-1339-z
  24. H. Wu, $L^p$ bounds for Marcinkiewicz integrals associated to surfaces of revolution, J. Math. Anal. Appl. (to appear)
  25. H. Wu, General Littlewood-Paley functions and singular integral operators on product spaces, Math. Nachr. 279 (2006), no. 4, 431-444 https://doi.org/10.1002/mana.200310369
  26. Y. Ying, Investigations on some operators with rough kernels in harmonic anal- ysis, Ph. D. Thesis (in Chinese), Zhejiang Univ., Hangzhou, 2002

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