Rough Marcinkiewicz integrals along certain smooth curves
Rough Marcinkiewicz integrals along certain smooth curves
Bolin MA1, Huoxiong WU2(), Xiating ZHAO2,3
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
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 Fα (Sm-1 × Sn-1) of the kernel implies the Lp-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.
. Rough Marcinkiewicz integrals along certain smooth curves[J]. Frontiers of Mathematics in China, 2012, 7(5): 857-872.
Bolin MA, Huoxiong WU, Xiating ZHAO. Rough Marcinkiewicz integrals along certain smooth curves. Front Math Chin, 2012, 7(5): 857-872.
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. Lp 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
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. Lp-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. Lp 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. Lp 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
