Marcinkiewicz integral with rough kernels

Shanzhen Lu

Front. Math. China ›› 2007, Vol. 3 ›› Issue (1) : 1-14.

PDF(187 KB)
PDF(187 KB)
Front. Math. China ›› 2007, Vol. 3 ›› Issue (1) : 1-14. DOI: 10.1007/s11464-008-0005-1
Survey Article

Marcinkiewicz integral with rough kernels

Author information +
History +

Abstract

In this summary paper, we will introduce some recent progress in the theory of Marcinkiewicz integral and will pay more attention to the case of rough kernels.

Keywords

Marcinkiewicz integral / rough kernel / commutator / Lp-boundedness / weighted Lp-boundedness / product space

Cite this article

Download citation ▾
Shanzhen Lu. Marcinkiewicz integral with rough kernels. Front. Math. China, 2007, 3(1): 1‒14 https://doi.org/10.1007/s11464-008-0005-1

References

[1.]
Al-Qassem H. Rough Marcinkiewicz integral operators on product spaces. Collect Math, 2005, 56: 275-297.
[2.]
Al-Salman A. Marcinkiewicz function along flat surfaces with Hardy space kernels. J Integral Equations Appl, 2005, 17: 357-373.
CrossRef Google scholar
[3.]
Al-Salman A., Al-Qassem H., Cheng L. Lp bounds for the function of Marcinkiewicz. Math Res Letter, 2002, 9: 697-700.
[4.]
Al-Salman A., Al-Qassem H., Cheng L. Marcinkiewicz integrals on product spaces. Studia Math, 2005, 167: 227-234.
CrossRef Google scholar
[5.]
Benedek A., Calderón A. P., Panzone R. Convolution operators on Banach value functions. Proc Nat Acad Sci, 1962, 48: 256-365.
[6.]
Calderón A. P., Zygmund A. On singular integrals with variable kernels. Applicable Anal, 1977, 7: 221-238.
CrossRef Google scholar
[7.]
Chen J., Ding Y., Fan D. Lp boundedness of rough Marcinkiewicz integral on product domains. Chinese J Contem Math, 2000, 49: 481-490.
[8.]
Chen J., Fan D., Ying Y. Rough Marcinkiewicz integrals with L(log+L)2 kernels on product spaces. Adv Math (China), 2001, 30: 179-181.
[9.]
Chen J., Fan D., Ying Y. The method of rotation and Marcinkiewicz integrals on product domains. Studia Math, 2002, 153: 41-58.
[10.]
Ding Y., Fan D., Pan Y. Weighted boundedness for a class of Marcinkiewicz integrals. Indiana Univ Math J, 1999, 48: 1037-1055.
CrossRef Google scholar
[11.]
Ding Y., Fan D., Pan Y. Lp boundedness of Marcinkiewicz with Hardy space function kernels. Acta Math Sinica (English Ser), 2000, 16: 593-600.
[12.]
Ding Y., Fan D., Pan Y. On Littlewood-Paley functions and singular integrals. Hokkaido Math J, 2000, 29: 537-552.
[13.]
Ding Y., Lin C., Shao S. On the Marcinkiewicz integral with variable kernels. Indiana Univ Math J, 2004, 53(3): 805-821.
CrossRef Google scholar
[14.]
Ding Y., Lu S., Xue Q. On Marcinkiewicz integral with homogeneous kernel. J Math Anal Appl, 2000, 245: 471-488.
CrossRef Google scholar
[15.]
Ding Y., Lu S., Xue Q. Marcinkiewicz integral on Hardy spaces. Integ Equa Oper Theory, 2002, 42: 174-182.
CrossRef Google scholar
[16.]
Ding Y., Lu S., Yabuta K. A problem on rough parametric Marcinkiewicz functions. J Austral Math Soc, 2002, 72: 13-21.
[17.]
Ding Y., Lu S., Yabuta K. On commutators of Marcinkiewicz integrals with rough kernel. J Math Anal Appl, 2002, 275: 60-68.
CrossRef Google scholar
[18.]
Ding Y., Lu S., Zhang P. Weighted weak type estimates for commutators of the Marcinkiewicz integrals. Sci China (Ser A-Math), 2004, 47: 83-95.
CrossRef Google scholar
[19.]
Duoandikoetxea J., Seijo E. Weighted inequalities for rough square functions through extrapolation. Studia Math, 2002, 149: 239-252.
[20.]
Fan D., Sato S. Weak type (1,1) estimates for Marcinkiewicz integrals with rough kernels. Tôhoku Math J, 2001, 53: 265-284.
CrossRef Google scholar
[21.]
Fefferman R., Soria F. The space weak H1. Studia Math, 1987, 85: 1-16.
[22.]
Han Y. Some properties of the S-function and Marcinkiewicz integrals. J Peking Univ, 1987, 5: 21-34.
[23.]
Hu G., Lu S., Yang D. Boundedness of rough singular integral operators on homogeneous Herz space. J Austral Math Soc (Ser A), 1999, 66: 201-223.
CrossRef Google scholar
[24.]
Hu G., Lu S., Yan D. Lp(ℝm × ℝn) boundedness for the Marcinkiewicz integral on product spaces. Sci China (Ser A-Math), 2003, 46: 75-82.
CrossRef Google scholar
[25.]
Hörmander L. Translation invariant operators. Acta Math, 1960, 104: 93-139.
CrossRef Google scholar
[26.]
Jiang Y., Lu S. A class of singular integral operators with rough kernel on product domains. Hokkaido Math J, 1995, 24: 1-7.
[27.]
Li J. Boundedness of some operators and their commutators. Ph D Thesis, 2005, Beijing: Beijing Normal University.
[28.]
Li R. Parameterized Marcinkiewicz integral with variable kernels. Master D Dissertation, 2006, Beijing: Beijing Normal University.
[29.]
Lu S., Mo H. The Boundedness of commutators for the Marcinkiewicz Integrals. Acta Math Sinica (Chinese Ser), 2006, 49: 481-490.
[30.]
Lu S., Mo H. Boundedness of higher order Marcinkiewicz type integrals. Inter J Math Math Sci, Article Id, 2006, 31705: 1-21.
CrossRef Google scholar
[31.]
Marcinkiewicz J. Sur quelques integrales de type de Dini. Annales de la Société Polon, 1938, 17: 42-50.
[32.]
Qiu S. Boundedness of Littlewood-Paley operators and Marcinkiewicz integrals. J Math Res Exposition, 1992, 12: 42-50.
[33.]
Sato S. Remarks on square functions in the Littlewood-Paley Theory. Bull Austral Math Soc, 1998, 58: 199-211.
[34.]
Stein E. M. On the function of Littlewood-Paley, Lusin and Marcinkiewicz. Trans Amer Math Soc, 1958, 88: 430-466.
CrossRef Google scholar
[35.]
Tao G. Boundedness of some operators on Lesbesgue spaces and Herz spaces. Master D Dissertation, 2002, Beijing: Beijing Normal University.
[36.]
Torchinsky A., Wang S. A note on the Marcinkiewicz integral. Colloq Math, 1990, 60/61: 235-243.
[37.]
Wang S. Some properties of Littlewood-Paley’s g-functions. Sci China (Ser AMath), 1985, 3: 252-262.
[38.]
Walsh T. On the function of Marcinkiewicz. Studia Math, 1972, 44: 203-217.
[39.]
Yabuta K. Boundedness of Littlewood operators. Math Japonica, 1996, 43: 143-150.
[40.]
Zygmund A. On certain integrals. Trans Amer Math Soc, 1944, 55: 170-204.
CrossRef Google scholar
AI Summary AI Mindmap
PDF(187 KB)

Accesses

Citations

Detail

Sections
Recommended

/