Boundedness of the Marcinkiewicz integrals with rough kernel associated to surfaces
Yong Ding, Qingying Xue, and Kôzô Yabuta
Source: Tohoku Math. J. (2) Volume 62, Number 2
(2010), 233-262.
Abstract
In this paper, the authors discuss the weighted $L^p$ boundedness for the rough Marcinkiewicz integrals associated to surfaces. More precisely, the kernel of our operator lacks smoothness not only on the unit sphere, but also in the radial directions. Moreover, the surface is defined by using a differentiable function with monotonicity and some properties on the positive real line. The results given in this paper improve and extend some known results.
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Permanent link to this document: http://projecteuclid.org/euclid.tmj/1277298647
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References
H. M. Al-Qassem, $L^p$ estimates for rough parametric Marcinkiewicz integrals, SUT J. Math. 40 (2004), 117--131.
Mathematical Reviews (MathSciNet): MR2123967
Zentralblatt MATH: 1069.42010
H. M. Al-Qassem, On weighted inequalities for parametric Marcinkiewicz integrals, J. Inequal. Appl. 2006, Article ID 91541, 17 pages.
Mathematical Reviews (MathSciNet): MR2215486
Zentralblatt MATH: 1100.42009
H. M. Al-Qassem, Weighted $L^p$ boundedness for the function of Marcinkiewicz, Kyungpook Math. J. 46 (2006), 31--48.
Mathematical Reviews (MathSciNet): MR2214798
Zentralblatt MATH: 1129.42377
H. M. Al-Qassem and Y. Pan, On rough maximal operators and Marcinkiewicz integrals along submanifolds, Studia Math. 190 (2009), 73--98.
Mathematical Reviews (MathSciNet): MR2457287
Zentralblatt MATH: 1165.42003
Digital Object Identifier: doi:10.4064/sm190-1-3
A. Al-Salman, H. Al-Qassem, L. Cheng and Y. Pan, $L^p$ bounds for the function of Marcinkiewicz, Math. Res. Lett. 9 (2002), 697--700.
Mathematical Reviews (MathSciNet): MR1906071
Zentralblatt MATH: 1035.42006
A. Benedek, A. P. Calderón and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356--365.
Mathematical Reviews (MathSciNet): MR133653
Zentralblatt MATH: 0103.33402
Digital Object Identifier: doi:10.1073/pnas.48.3.356
JSTOR: links.jstor.org
L. Colzani, Hardy spaces on sphere, Ph. D. Thesis, Washington University, St. Louis, MO, 1982.
L. Colzani, M. Taibleson and G. Weiss, Maximal estimates for Cesàro and Riesz means on sphere, Indiana Univ. Math. J. 33 (1984), 873--889.
Mathematical Reviews (MathSciNet): MR763947
Digital Object Identifier: doi:10.1512/iumj.1984.33.33047
Y. Ding, D. Fan and Y. Pan, Weighted boundedness for a class of rough Marcinkiewicz integrals, Indiana Univ. Math. J. 48 (1999), 1037--1055.
Mathematical Reviews (MathSciNet): MR1736970
Zentralblatt MATH: 0949.42014
Digital Object Identifier: doi:10.1512/iumj.1999.48.1696
Y. Ding, D. Fan and Y. Pan, $L^p$-boundedness of Marcinkiewicz integrals with Hardy space function kernel, Acta Math. Sinica (English Ser.) 16 (2000), 593--600.
Mathematical Reviews (MathSciNet): MR1813455
Zentralblatt MATH: 0966.42008
Digital Object Identifier: doi:10.1007/s101140000015
Y. Ding, D. Fan and Y. Pan, On the $L^p$ boundedness of Marcinkiewicz Integrals, Michigan Math. J. 50 (2002), 17--26.
Mathematical Reviews (MathSciNet): MR1897030
Digital Object Identifier: doi:10.1307/mmj/1022636747
Project Euclid: euclid.mmj/1022636747
Y. Ding and S. Lu, Weighted $L^p$-boundedness for higher order commutators of oscillatory singular integrals, Tohoku Math. J. 48 (1996), 437--449.
Mathematical Reviews (MathSciNet): MR1404513
Digital Object Identifier: doi:10.2748/tmj/1178225342
Project Euclid: euclid.tmj/1178225342
J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), 869--880.
Mathematical Reviews (MathSciNet): MR1089418
Zentralblatt MATH: 0770.42011
Digital Object Identifier: doi:10.2307/2154381
J. Duoandikoetxea and J. L. RubiodeFrancia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541--561.
Mathematical Reviews (MathSciNet): MR837527
Zentralblatt MATH: 0568.42012
Digital Object Identifier: doi:10.1007/BF01388746
D. Fan and Y. Pan, A singular integral operators with rough kernel, Proc. Amer. Math. Soc. 125 (1997), 3695--3703.
Mathematical Reviews (MathSciNet): MR1422868
Zentralblatt MATH: 0887.42009
Digital Object Identifier: doi:10.1090/S0002-9939-97-04111-7
JSTOR: links.jstor.org
D. Fan, Y. Pan and D. Yang, A weighted norm inequality for rough singular integrals, Tohoku Math. J. 51 (1999), 141--161.
Mathematical Reviews (MathSciNet): MR1690023
Digital Object Identifier: doi:10.2748/tmj/1178224808
Project Euclid: euclid.tmj/1178224808
L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math. 104 (1960), 93--140.
Mathematical Reviews (MathSciNet): MR121655
Zentralblatt MATH: 0093.11402
Digital Object Identifier: doi:10.1007/BF02547187
M.-Y. Lee and C.-C. Lin, Weighted $L^p$ boundedness of Marcinkiewicz integral, Integral Equations Operator Theory 49 (2004), 211--220.
Mathematical Reviews (MathSciNet): MR2060372
Zentralblatt MATH: 1050.42015
Digital Object Identifier: doi:10.1007/s00020-002-1204-x
S. Lu, Y. Ding and D. Yan, Singular integrals and related topics, World Scientific, Singapore, 2007.
Mathematical Reviews (MathSciNet): MR2354214
Zentralblatt MATH: 1124.42011
M. Sakamoto and K. Yabuta, Boundedness of Marcinkiewicz functions, Studia Math. 135 (1999), 103--142.
Mathematical Reviews (MathSciNet): MR1690749
Zentralblatt MATH: 0930.42009
E. M. Stein, On the functions of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430--466.
Mathematical Reviews (MathSciNet): MR112932
Zentralblatt MATH: 0105.05104
Digital Object Identifier: doi:10.2307/1993226
JSTOR: links.jstor.org
E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proceedings of the International Congress of Mathematicians, Vol. 1.2 (Berkeley, Calif., 1986), 196--221, Amer. math. Soc., Providence, RI, 1987.
Mathematical Reviews (MathSciNet): MR934224
Zentralblatt MATH: 0718.42012
Q. Xue and K. Yabuta, $L^2$-boundedness of Marcinkiewicz integrals along surfaces with variable kernels, Sci. Math. Jpn. 63 (2006), 269--282.
Mathematical Reviews (MathSciNet): MR2221445
Q. Xue and K. Yabuta, Correction and Addition to: “$L^2$-boundedness of Marcinkiewicz integrals along surfaces with variable kernels”, Sci. Math. Jpn. 65 (2007), 291--298.
Mathematical Reviews (MathSciNet): MR2300350
X. Ye and X. Zhu, A note on certain block spaces on the unit sphere, Acta Math. Sinica (E.S.) 22 (2006), 1843--1846.
Mathematical Reviews (MathSciNet): MR2262445
Zentralblatt MATH: 1157.42306
Digital Object Identifier: doi:10.1007/s10114-005-0781-4
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