ROUGH MULTIPLE SINGULAR INTEGRALS ALONG HYPERSURFACES

doi:10.1016/S0252-9602(11)60383-4

数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (5): 2081-2098.doi: 10.1016/S0252-9602(11)60383-4

ROUGH MULTIPLE SINGULAR INTEGRALS ALONG HYPERSURFACES

邓浏睿|李中凯|马柏林|伍火熊^*

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China； School of Mathematical Sciences, Capital Normal University, Beijing 100048, China； College of Science and Information Engineering, Jiaxing University, Jiaxing 314001, China； School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

收稿日期:2009-02-10 修回日期:2011-04-15 出版日期:2011-09-20 发布日期:2011-09-20
通讯作者: 伍火熊，huoxwu@xmu.edu.cn E-mail:dengzimu@yahoo.com.cn；lizk@mail.cnu.edu.cn；blma@mail.zjxu.edu.cn；huoxwu@xmu.edu.cn
基金资助:
Supported by the NSFC (10771054, 10971141, 11071200) and the NFS of Beijing (1092004) and the NFS of Fujian Province (2010J01013).

ROUGH MULTIPLE SINGULAR INTEGRALS ALONG HYPERSURFACES

DENG Liu-Rui, LI Zhong-Kai, MA Bo-Lin, WU Huo-Xiong^*

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China； School of Mathematical Sciences, Capital Normal University, Beijing 100048, China； College of Science and Information Engineering, Jiaxing University, Jiaxing 314001, China； School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received:2009-02-10 Revised:2011-04-15 Online:2011-09-20 Published:2011-09-20
Contact: WU Huo-Xiong，huoxwu@xmu.edu.cn E-mail:dengzimu@yahoo.com.cn；lizk@mail.cnu.edu.cn；blma@mail.zjxu.edu.cn；huoxwu@xmu.edu.cn
Supported by:
Supported by the NSFC (10771054, 10971141, 11071200) and the NFS of Beijing (1092004) and the NFS of Fujian Province (2010J01013).

摘要/Abstract

摘要：

In this paper, the authors study the mapping properties of singular integrals on product domains with kernels in L(log+L)^ε(S^m⁻¹× Sⁿ⁻¹) (ε = 1 or 2) supported by hyper-surfaces. The L^p bounds for such singular integral operators as well as the related
Marcinkiewicz integral operators are established, provided that the lower dimensional max-imal function is bounded on L^q(R³) for all q > 1. The condition on the integral kernels is known to be optimal.

关键词: multiple singular integral, Marcinkiewicz integral, maximal function, hyper-surface, rough kernel

Abstract:

Key words: multiple singular integral, Marcinkiewicz integral, maximal function, hyper-surface, rough kernel

中图分类号:

42B20

邓浏睿, 李中凯, 马柏林, 伍火熊. ROUGH MULTIPLE SINGULAR INTEGRALS ALONG HYPERSURFACES[J]. 数学物理学报(英文版), 2011, 31(5): 2081-2098.

DENG Liu-Rui, LI Zhong-Kai, MA Bo-Lin, WU Huo-Xiong. ROUGH MULTIPLE SINGULAR INTEGRALS ALONG HYPERSURFACES[J]. Acta mathematica scientia,Series B, 2011, 31(5): 2081-2098.

参考文献

[1] Al-Salman A. Maximal operators with rough kernels on product domains. J Math Anal Appl, 2006, 311: 338–351

[2] Al-Qassem H, Al-Salman A, Cheng L, Pan Y. Marcinkiewicz integrals on product domains. Studia Math, 2006, 167: 227–234

[3] Al-Salman A, Al-Qassem H, Pan Y. Singular integrals on product domains. Indiana Univ Math J, 2006, 55(1): 369–387

[4] Al-Salman A, Pan Y. Singular integrals with rough kernels in L(log+L)(Sn−1). J London Math Soc, 2002, 66(2): 153–174

[5] Al-Qassem H, Pan Y. Singular integrals along surfaces of revolution with rough kernels. SUT J Math, 2003, 39(1): 55–70

[6] Chen L, Fan D. On singular integrals along surfaces related to block spaces. Integr Equ Oper Theory, 1997, 29: 261–268

[7] Cheng L C, Pan Y. Lp bounds for singular integrals associated to surfaces of revolution. J Math Anal Appl, 2002, 265: 163–169

[8] Ding Y, Fan D, Pan Y. Marcinkiewicz integrals with rough kernels on product spaces. Yokohama Math J, 2001, 49: 1–15

[9] Duoandikoetxea J. Multiple singular integrals and maximal functions along hypersurfaces. Ann Inst Four-ier, 1986, 36: 185–206

[10] Fan D, Guo K, Pan Y. Singular integrals with rough kernels on product spaces. Hokkaido Math J, 1999, 28: 435–460

[11] Fan D, Pan Y. Singular integral operators with rough kernels supported by subvarieties. Amer J Math, 1997, 119: 799–839

[12] Fefferman R. Singular integrals on product domains. Bull Amer Math Soc, 1981, 4: 195–201

[13] Fefferman R, Stein E M. Singular integrals on product domains. Adv Math, 1982, 45: 117–143

[14] Kim W, Wainger S, Wright J, Ziesler S. Singular integrals and maximal functions associated to surfaces of revolution. Bull London Math Soc, 1996, 28: 291–296

[15] Li J. Boundedness of Some Operators and Commutators [D]. Beijing: Beijing Normal Univ, 2005 (in Chinese)

[16] Lu S, Pan Y, Yang D. Rough singular integrals associated to surfaces of revolution. Proc Amer Math Soc, 2001, 129: 2931–2940

[17] Nagel A, Wainger S. L2-boundedness of Hilbert transforms along surfaces and convolution operators ho-mogeneous with respect to a multiple parameter group. Amer J Math, 1977, 99: 761–785

[18] Pan Y, Tang L, Yang D. Boundedness of rough singular integrals associated with surfaces of revolution. Adv Math (China), 2003, 32(2): 677–682

[19] Ricci F, Stein E M. Multiparameter singular integrals and maximal functions. Ann Inst Fourier, 1992, 42: 637–670

[20] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Prince-ton, New Jersey: Princeton Univ Press, 1993

[21] Stein E M, Wainger S. Problems in harmonic analysis related to curvature. Bull Amer Math Soc, 1978, 84: 1239–1295

[22] Seeger A, Wainger S, Wright J, Ziesler S. Singular integral and maximal operators associated to hypersur-faces: Lp theory. J Geom Anal, 2005, 15(3): 477–498

[23] Strichartz R S. Singular integrals supported in manifolds. Studia Math, 1982, 74: 137–151

[24] Vance J T. Lp-boundedness of the multiple Hilbert transform along a surface. Pacific J Math 1983, 108: 221–241

[25] Wang M, Chen J, Fan D. A note on certain square functions on product spaces. Sci China, Ser A, 2005, 35(4): 1–8

[26] Wu H. General Littlewood-Paley functions and singular integral operators on product spaces. Math Nachr, 2006, 279(4): 431–444

[27] Wu H. A rough multiple Marcinkiewicz integral along continuous surface. Tohoku Math J, 2007, 59: 145–166

[28] Wu H, Xu J. Rough Marciniewicz integrals associated to surfaces of revolution on product domains. Acta Math Sci, 2009, 29B(2): 294–304

ROUGH MULTIPLE SINGULAR INTEGRALS ALONG HYPERSURFACES

ROUGH MULTIPLE SINGULAR INTEGRALS ALONG HYPERSURFACES

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