沿Van der Coupt型曲面的粗糙核参数型Marcinkiewicz积分算子的有界性

数学物理学报 ›› 2017, Vol. 37 ›› Issue (1): 38-53.

沿Van der Coupt型曲面的粗糙核参数型Marcinkiewicz积分算子的有界性

刘风¹, 徐秀荣², 张婧³

1. 山东科技大学数学与系统科学学院山东青岛 266590;
2. 宿州学院数学与统计学院安徽宿州 234000;
3. 伊犁师范学院数学与统计学院新疆伊宁 835000

收稿日期:2016-07-09 修回日期:2016-12-04 出版日期:2017-02-26 发布日期:2017-02-26
通讯作者: 张婧 E-mail:zjmath66@126.com
作者简介:刘风,E-mail:liufeng860314@163.com;徐秀荣,E-mail:xxr9336@163.com
基金资助:
新疆维吾尔自治区自然科学基金（2016D01C381）

Rough Marcinkiewicz Integrals Associated with Surfaces of Van der Corput Type

Liu Feng¹, Xu Xiurong², Zhang Jing³

1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Shandong Qingdao 266590;
2. School of Mathematics and Statistics, Suzhou University, Anhui Suzhou 234000;
3. School of Mathematics and Statistics, Yili Normal College, Xinjiang Yining 835000

Received:2016-07-09 Revised:2016-12-04 Online:2017-02-26 Published:2017-02-26
Supported by:
Supported by the Natural Science Foundation of Xinjiang Province (2016D01C381)

摘要/Abstract

摘要：

该文研究了沿Van der Corput型曲面的粗糙核参数型Marcinkiewicz积分算子.通过外插技巧，在积分核满足相当弱的尺寸条件下，建立了这些算子的L^p有界性.这些结果在很大程度上推广并改善了已有的相关结论.

关键词: Marcinkiewicz积分, 粗糙核, Van der Corput型曲面, 外插

Abstract:

This paper is concerned with the parametric Marcinkiewicz integrals with rough kernels associated with surfaces of Van der Corput type. Under the rather weakened size conditions on the integral kernels both on the unit sphere and in the radial direction, the L^p bounds of such operators are given in an extrapolation argument. Some previous results are greatly extended and improved.

Key words: Marcinkiewicz integrals, Rough kernels, Surfaces of Van der Corput type, Extrapolation

中图分类号:

O174.3

刘风, 徐秀荣, 张婧. 沿Van der Coupt型曲面的粗糙核参数型Marcinkiewicz积分算子的有界性[J]. 数学物理学报, 2017, 37(1): 38-53.

Liu Feng, Xu Xiurong, Zhang Jing. Rough Marcinkiewicz Integrals Associated with Surfaces of Van der Corput Type[J]. Acta mathematica scientia,Series A, 2017, 37(1): 38-53.

参考文献

[1] Al-Salman A. Maximal functions along surfaces on product domains. Anal Math, 2008, 34:163-175
[2] Al-Salman A. On Marcinkiewicz integrals along flat surfaces. Turk J Math, 2005, 29:111-120
[3] Al-Salman A, Al-Qassem H. Flat Marcinkiewicz integral operators. Turk J Math, 2002, 26:329-338
[4] Al-Salman A, Al-Qassem H, Cheng L C, Pan Y B. L^p bounds for the function of Marcinkiewicz. Math Res Lett, 2002, 9:697-700
[5] Al-Salman A, Pan Y B. Singular integrals with rough kernels in Llog⁺L(S^n-1). J London Math Soc, 2002, 66(2):153-174
[6] Al-Salman H, Pan Y B. On certain estimates for Marcinkiewicz integrals and extrapolation. Collect Math, 2009, 60(2):123-145
[7] Benedek A, Calderón A, Panzone R. Convolution operators on Banach space valued functions. Proc Natl Acad Sci USA, 1962, 48:356-365
[8] Chen J C, Fan D S, Pan Y B. A note on a Marcinkiewicz integral operator. Math Nachr, 2001, 227:33-42
[9] Duoandikoetxea J, Rubio de Francia J L. Maximal and singular integral operators via Fourier transform estimates. Invent Math, 1986, 84:541-561
[10] Ding Y, Fan D S, Pan Y B. L^p-boundedness of Marcinkiewicz integrals with Hardy space function kernel. Acta Math Sin (Engl Ser), 2000, 16:593-600
[11] Ding Y, Lu S Z, Yabuta K. A problem on rough parametric Marcinkiewicz functions. J Aust Math Soc, 2002, 72:13-21
[12] Ding Y, Xue Q Y, Yabuta K. Boundedness of the Marcinkiewicz integrals with rough kernel associated to surfaces. Tohoku Math J, 2010, 62:233-262
[13] Ding Y, Pan Y B. L^p bounds for Marcinkiewicz integrals. Proc Edin Math Soc, 2003, 46:669-677
[14] Fan D S, Pan Y B. Singular integral operators with rough kernels supported by subvarieties. Amer J Math, 1997, 119:799-839
[15] 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
[16] Hörmander L. Estimates for translation invariant operators in L^p spaces. Acta Math, 1960, 104:93-140
[17] Liu F, Zhang D Q. Parabolic Marcinkiewicz integrals associated to polynomials compound curves and extrapolation. Bull Korean Math Soc, 2015, 52(3):771-788
[18] Sakamoto M, Yabuta K. Boundedness of Marcinkiewicz functions. Studia Math, 1999, 135:103-142
[19] Sato S. Estimates for singular integrals along surfaces of revolution. J Aust Math Soc, 2009, 86:413-430
[20] Sato S. Estimates for singular integrals and extrapolation. Studia Math, 2009, 192:219-233
[21] Stein E M. On the function of Littlewood-Paley, Lusin and Marcinkiewicz. Trans Amer Math Soc, 1958, 88:430-466
[22] Stein E M. Harmonic Analysis:Real-Variable Methods, Orthogonality and Oscillatory Integral. Princeton:Princeton University Press, 1993
[23] Walsh T. On the function of Marcinkiewic. Studia Math, 1972, 44:203-217
[24] Wu H X. On Marcinkiewicz integral operators with rough kernels. Integ Equa Oper Theory, 2005, 52:285-298
[25] Wu H X. L^p bounds for Marcinkiewicz integrals associates to surfaces of revolution. J Math Anal Appl, 2006, 321:811-827
[26] Wu H X, Xu J K. Rough Marcinkiewicz integrals associated to surfaces of revolution on product domains. Acta Math Sci, 2009, 29B(2):294-304
[27] Xu H, Chen J C, Ying Y M. A note on Marcinkiewicz integrals with H^1 kernels. Acta Math Sci, 2003, 23B(1):133-138