Rough Marcinkiewicz integrals along certain smooth curves

doi:10.1007/s11464-012-0237-y

Front Math Chin

2012, Vol. 7

Issue (5) : 857-872 https://doi.org/10.1007/s11464-012-0237-y

RESEARCH ARTICLE

Rough Marcinkiewicz integrals along certain smooth curves

Bolin MA¹, Huoxiong WU²(

), Xiating ZHAO^2,3

1. College of Mathematics and Information Engineering, Jiaxing University, Jiaxing 314001, China; 2. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China; 3. Third Middle-School of Zhuhai City, Zhuhai 519000, China

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Abstract

This paper is devoted to the study of the multiple-parameter rough Marcinkiewicz integral operators associated with certain smooth curves. It is shown that the Grafakos-Stefanov type size condition F_α (S^m-1 × S^n-1) of the kernel implies the L^p-boundedness of these Marcinkiewicz integral operators for some α>1/2 and 1+12α<p<1+2α, which is an essential improvement of certain previous results.

Keywords Marcinkiewicz integral rough kernel smooth curve LittlewoodPaley theory Fourier transform estimate

Corresponding Author(s): WU Huoxiong,Email:huoxwu@xmu.edu.cn

Issue Date: 01 October 2012

Cite this article:

Bolin MA,Huoxiong WU,Xiating ZHAO. Rough Marcinkiewicz integrals along certain smooth curves[J]. Front Math Chin, 2012, 7(5): 857-872.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-012-0237-y
https://academic.hep.com.cn/fmc/EN/Y2012/V7/I5/857

1	Al-Qassem H, Al-Salman A, Cheng L, Pan Y. Marcinkiewicz integrals on product domains. Studia Math , 2005, 167: 227-234 doi: 10.4064/sm167-3-4
2	Al-Salman A. On Marcinkiewicz integrals along flat surfaces. Turk J Math , 2005, 29: 111-120
3	Al-Salman A. Rough Marcinkiewicz integrals on product spaces. Int Math Forum , 2007, 23(2): 1119-1128
4	Al-Salman A, Al-Qassem H. L^p bounds for the function of Marcinkiewicz. Math Res Lett , 2002, 9: 697-700
5	Benedek A, Calderón A P, Panzone R. Convolution operators on Banach space valued functions. Proc Nat Acad Sci USA , 1962, 48: 356-365 doi: 10.1073/pnas.48.3.356
6	Chen J, Fan D, Pan Y. A note on a Marcinkiewicz integral operator. Math Nachr , 2001, 227: 33-42 doi: 10.1002/1522-2616(200107)227:1<33::AID-MANA33>3.0.CO;2-0
7	Chen J, Fan D, Ying Y. Rough Marcinkiewicz integrals with L(log L)² kernels. Adv Math (China) , 2001, 30: 179-181
8	Choi Y. Marcinkiewicz integrals with rough homogeneous kernel of degree zero in product domains. J Math Anal Appl , 2001, 261: 53-60 doi: 10.1006/jmaa.2001.7465
9	Ding Y. L²-boundedness of Marcinkiewicz integral with rough kernel. Hokkaido Math J , 1998, 27: 105-115
10	Ding Y, Fan D, Pan Y. L^p-boundedness of Marcinkiewicz integrals with Hardy space function kernels. Acta Math Sin (Engl Ser) , 200, 16: 593-600
11	Duoandikoetxea J, Rubio de Francia J L. Maximal and singular integral operators via Fourier transform estimates. Invent Math , 1986, 84: 541-561 doi: 10.1007/BF01388746
12	Grafakos L, Stefanov A. L^p bounds for singular integrals and maximal singular integrals with rough kernels. Indiana Univ Math J , 1998, 47: 455-469 doi: 10.1512/iumj.1998.47.1521
13	Hu G, Lu S, Yan D. Lp(?m×?n) boundedness for the Marcinkiewicz integral on product spaces. Sci China, Ser A , 2003, 46(1): 75-82 doi: 10.1360/03ys9008
14	Stein E M. On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans Amer Math Soc , 1958, 88: 430-466 doi: 10.1090/S0002-9947-1958-0112932-2
15	Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton University Press, 1993, 332
16	Wu H. On Marcinkiewicz integral operators with rough kernels. Integral Equations Operator Theory , 2005, 52: 285-298 doi: 10.1007/s00020-004-1339-z
17	Wu H. L^p bounds for Marcinkiewicz integrals associated to surfaces of revolution. J Math Anal Appl , 2006, 321(2): 811-827 doi: 10.1016/j.jmaa.2005.08.087
18	18. Wu H. Boundedness of multiple Marcinkiewicz integral operators with rough kernels. J Korean Math Soc , 2006, 43(3): 635-658 doi: 10.4134/JKMS.2006.43.3.635
19	Wu H. General Littlewood-Paley functions and singular integral operators on product spaces. Math Nachr , 2006, 279(4): 431-444 doi: 10.1002/mana.200310369
20	Wu H. A rough multiple Marcinkiewicz integral along continuous surfaces. Tohoku Math J , 2007, 59(2): 145-166 doi: 10.2748/tmj/1182180732

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