Acta mathematica scientia, Series B >
ROUGH MARCINKIEWICZ INTEGRALS ASSOCIATED TO SURFACES OF REVOLUTION ON PRODUCT DOMAINS
Received date: 2006-11-19
Online published: 2009-03-20
Supported by
Supported by the NSF of China (G10571122) and the NFS of Fujian Province of China (Z0511004)
In this article, certain Marcinkiewicz integral operators associated to surfaces of revolution on product domains were studied. The Lp boundedness for these operators are established under some rather weak conditions on kernels. The main results essentially improve and extend some known results.
Wu Huoxiong , Xu Jiankai . ROUGH MARCINKIEWICZ INTEGRALS ASSOCIATED TO SURFACES OF REVOLUTION ON PRODUCT DOMAINS[J]. Acta mathematica scientia, Series B, 2009 , 29(2) : 294 -304 . DOI: 10.1016/S0252-9602(09)60030-8
[1] Al-Qassem H. Singular integrals along surfaces on product domains. Anal Theory and Appl, 2004, 22(2):
99–112
[2] Al-Salman A. Flat singular integrals on product domains. Filomat (Nis), 2004, 18: 1–13
[3] Al-Salman A, Al-Qassem H, Cheng L C, Pan Y. Marcinkiewicz integrals on product spaces. Studia Math,
2005, 167: 227–234
[4] Chen J, Ding Y, Fan D. Lp boundedness of the rough Marcinkiewicz integral on product domains. Chinese
J Contemp Math, 2000, 21(1): 47–54
[5] Chen J, Fan D, Ying Y. Rough Marcinkiewicz integrals with L(log+L)2 kernels on product spaces. Adv
Math (China), 2001, 30(2): 5–18
[6] Carbery A, Christ M, Wright J. Multidimensional van der Corput and sublevel set estimates. J Amer
Math Soc, 1999, 12: 981–1015
[7] Cheng L, Pan Y. Lp bounds for singular integrals associated to surfaces of revolution. J Math Anal Appl,
1998, 27: 105–115
[8] Chanillo S, Wheeden R L. Inequalities for Peano maximal functions and Marcinkiewicz integrals. Duke
Math J, 1983, 50: 573-603
[9] Chanillo S, Wheeden R L. Relations between Peano derivatives and Marcinkiewicz integrals. In: Conference
on Harmonic Analysis in Honor of Antoni Zygmund, Vols. I, II, Wadsworth, 1983. 508–525.
[10] Ding Y. L2-boundedness of Marcinkiewicz integral with rough kernel. Hokkaido Math J, 1998, 27(1):
105–115
[11] Ding Y, Fan D, Pan Y. On the Lp-boundedness of Marcinkiewicz integrals. Michigan Math J, 2002, 50:
17–26
[12] Ding Y, Pan Y. Lp bounds for Marcinkiewicz integrals. Proc Edindurg Math Soc, 2003, 46: 669–677
[]3] Duoandikoetxea J. Multiple singular integrals and maximal functions along hypersurfaces. Ann Inst
Fourier, Gronble, 1986, 36(4): 185–206
[14] Duoandikoetxea J, Rubio de Francia J L. Maximal and singular integral operators via Fourier transform
estimates. Invent Math, 1986, 84: 541–561
[15] Grafakos L, Stefanov A. Lp bounds for singular integrals and maximal singular integrals with rough kernels.
Indiana Univ Math J, 1998, 47: 455–469
[]6] Hu G, Lu S, Yan D. Lp(Rm × Rn) boundedness of Marcinkiewicz integral on product spaces. Sci China,
Ser A, 2003, 46(1): 75–82
[17] Hörmander L. Estimates for translation invariant operators in Lp spaces. Acta Math, 1960, 104: 93–139
[18] Pan Y, Tang L, Yang D. Boundedness of rough singular integrals associated with surfaces of revolution.
Adv Math (China), 2003, 32(6): 677–682
[19] Sakamoto M, Yabuta K. Boundedness of Marcinkiewicz functions. Studia Math, 1999, 135: 103–142
[20] Stein E M. On the function of Littlewood-Paley, Lusin and Marcinkiewicz. Trans Amer Math Soc, 1958,
88(2): 199–211
[21] Stein E M. Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integral. Princeton,
NJ: Princeton Univ Press, 1993
[22] Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton, New Jersey: Princeton
Univ Press, 1970
[23] Walsh T. On the function of Marcinkiewicz. Studia Math, 1972, 44: 203–217
[24] Wu H. Lp bounds for Marcinkiewicz integrals associated to surfaces of revolution. J Math Anal Appl,
2006, 321: 811-827
[25] Wu H. General Littlewood-Paley functions and singular integral operators on product spaces. Math Nachr,
2006, 279(4): 431–444
[26] Wu H. On multiple Marcinkiewicz integrals related to block spaces. Acta Math Sinica, 2005, 48(6):
1233–1250
[27] Wu H. Boundedness of multiple Marcinkiewicz integral operators with rough kernels. J Korean Math Soc,
2006, 43(3): 635–658
[28] Ying Y. Investigations on some operators with rough kernels in harmonic analysis
[Ph. D. Thesis (in
Chinese)]. Hangzhou: Zhejiang Univ, 2002
/
| 〈 |
|
〉 |