关于Marcinkiewicz型算子及其交换子的某些估计
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摘要
本文共三章,主要研究三个方面的内容:Marcinkiewicz积分在加权Campanato空间中的有界性;分数次型Marcinkiewicz积分的有界性;分数次型Marcinkiewicz积分交换子在Hardy空间中的估计.
行文结构安排如下:
第一章,得到了Marcinkiewicz积分在加权Campanato空间中的有界性,其中核满足.这推广了已知的结果.
设S~(n-1)表示Rn中的单位球面,Ω是S~(n-1)上具有消失性的零次齐次函数,即
对任意的x = 0 ,记x = x/|x|.此时高维的分数次型Marcinkiewicz积分定义为
其中
第二章,得到了分数次型Marcinkiewicz积分的有界性,其中核满足L~q-Dini条件.
设b(x)∈Lloc(Rn),则由μ_(Ω,α)和b生成的高维的分数次型Marcinkiewicz积分交换子定义为
其中
第三章,得到了由Lipα(0 <α≤1)函数和μ?,α生成的分数次型Marcinkiewicz积分交换子在Hardy空间中的有界性,其中的核满足某种对数型的Lipschitz条件.
There are three chapters in this thesis,which focuses on three contents:the bounded-ness of the Marcinkiewicz integral operator on certain weighted Campanato spaces; thebounededness of the fractional type Marcinkiewicz integral; estimates for the fractionaltype Marcinkiewicz Integral Commutator.
The outline of the paper is arranged as follows:
In Chapter 1,we obtain the boundedness of the Marcinkiewicz integral operator oncertain weighted Campanato spaces,which generalize the results that we know.
Suppose that S~(n-1) is the unit sphere of Rn(n≥2)equipped with the normalizedLebesgue measure dσ= dσ(x). LetΩbe homogeneous of degree zero and satisfies thecancelation conditionwhere x = x/|x| for any x = 0. We define the fractional type Marcinkiewicz integral ofhigher dimension bywhere
In chapter 2,we obtain the boundedness of the fractional type Marcinkiewicz Integralμ_(Ω,α) with the kernel satisfying L~q-Dini condition.Let b(x)∈Lloc(Rn), then the fractional type Marcinkiewicz integral commutator ofhigher dimension is defined bywhere
In chapter 3,we obtain estimates for the fractional type Marcinkiewicz Integral Com-mutatorμ_b~(Ω,α) on Hardy spaces with the kernel satisfying the logarithmic type Lipschitzcondition, where b∈Lip_α(0 <α≤1).
行文结构安排如下:
第一章,得到了Marcinkiewicz积分在加权Campanato空间中的有界性,其中核满足.这推广了已知的结果.
设S~(n-1)表示Rn中的单位球面,Ω是S~(n-1)上具有消失性的零次齐次函数,即
对任意的x = 0 ,记x = x/|x|.此时高维的分数次型Marcinkiewicz积分定义为
其中
第二章,得到了分数次型Marcinkiewicz积分的有界性,其中核满足L~q-Dini条件.
设b(x)∈Lloc(Rn),则由μ_(Ω,α)和b生成的高维的分数次型Marcinkiewicz积分交换子定义为
其中
第三章,得到了由Lipα(0 <α≤1)函数和μ?,α生成的分数次型Marcinkiewicz积分交换子在Hardy空间中的有界性,其中的核满足某种对数型的Lipschitz条件.
There are three chapters in this thesis,which focuses on three contents:the bounded-ness of the Marcinkiewicz integral operator on certain weighted Campanato spaces; thebounededness of the fractional type Marcinkiewicz integral; estimates for the fractionaltype Marcinkiewicz Integral Commutator.
The outline of the paper is arranged as follows:
In Chapter 1,we obtain the boundedness of the Marcinkiewicz integral operator oncertain weighted Campanato spaces,which generalize the results that we know.
Suppose that S~(n-1) is the unit sphere of Rn(n≥2)equipped with the normalizedLebesgue measure dσ= dσ(x). LetΩbe homogeneous of degree zero and satisfies thecancelation conditionwhere x = x/|x| for any x = 0. We define the fractional type Marcinkiewicz integral ofhigher dimension bywhere
In chapter 2,we obtain the boundedness of the fractional type Marcinkiewicz Integralμ_(Ω,α) with the kernel satisfying L~q-Dini condition.Let b(x)∈Lloc(Rn), then the fractional type Marcinkiewicz integral commutator ofhigher dimension is defined bywhere
In chapter 3,we obtain estimates for the fractional type Marcinkiewicz Integral Com-mutatorμ_b~(Ω,α) on Hardy spaces with the kernel satisfying the logarithmic type Lipschitzcondition, where b∈Lip_α(0 <α≤1).
引文
[1] E. Stein. On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. TransAmer Math Soc[J]. 1958, 88(2): 430-466.
[2] A. Benedek, A. Calder′on and R. Panzone. Convolution operators on Banach spacevalued functions. Proc Nat Acad Sci U.S.A.[J]. 1962, 48(3): 356-365.
[3] J.Chen, D. Fan and Y. Pan. A note on a Marcinkiewicz integral operator. MathNach[J]. 2001, 227(1): 33-42.
[4] Y. Ding, D. Fan and Y. Pan. Weighted boundedness for a class of rough Marcinkiewiczintegrals. Indiana Univ Math J[J]. 1999, 48(3): 1037-1055.
[5] Y. Ding, D. Fan and Y. Pan. LP-boundedness of Marcinkiewicz integrals withHardy space function kernels. Acta Math Sinica (English series)[J]. 2000, 16(4): 593-600.
[6] J. Chen, D. Fan and Y. Ying. The method of rotation and Marcinkiewicz integralson product domains. Studia Math[J]. 2002, 153(1): 41-58.
[7] J. Chen, D. Fan and Y. Ying. Rough Marcinkiewicz integrals withL(Log+L)2 kernelson product spaces. Adv Math(China)[J]. 2001, 30(2): 179-182.
[8] J. Chen, Y. Ding and D. Fan. Certain square functions on product spaces. MathNach[J]. 2001, 230(1): 5-18.
[9] Y. Ding, D. Fan and Y. Pan. Marcinkiewicz integrals with rough kernels on productspaces. Yokohama Math J[J]. 2001, 49: 1-15.
[10] A. Torchinsky and S. Wang. A note on Marcinkiewicz integral. Colloq Math[J]. 1990,60(2): 235-243.
[11] Y. Ding, S. Lu and K. Yabuta. On commutators of Marcinkiewicz integrals withrough kernel. Math Anal Appd[J]. 2002, 140: 60-68.
[12] Y. Han. Some properties of S-function and Marcinkiewicz integrals. Acta Sci NaturUniv pekingniss[J]. 1978, 5: 21-34.
[13] S. Qiu. Boundedness of Littlewood-paly operators and Marcinkiewicz integral onεα,p.Math Res Exposition[J]. 1992, 12: 41-50.
[14] Y. Wang. Littlewood-paly g- operator on weighted Campanato spaces. Journal ofHenan Normal University (Natural Seience)[J]. 1995, 23(3): 22-25.
[15] Y. Ding and S. Lu. Weighted norm inequalities for fractional integral operator withrough kernel. Can J Math[J]. 1998, 50(1): 29-39.
[16] Chanillo S.,Waton D. and Wheeden R.l. Some integral and maximal operator ralatedto starlike sets. Studia Math[J]. 1993, 107(1): 223-255.
[17] Y. Ding, S. Lu and Q. Xue. On Marcinkiewicz integral with homogeneous kernels. JMath Anal and Appl[J]. 2000, 245(2): 471-488.
[18] C. Perez. Endpoint estimates for commutator of singular integral operators. J FunctAnal[J]. 1995, 128: 163-185.
[19] S. Lu, and D. Yang. The local version of Hp spaces at the origin. studia Math[J].1995, 116: 103-131.
[20] S. Lu and D. Yang. The weighted Herz-type Hardy spaces and its applications. Sciencein China(Ser. A)(English)[J]. 1995, 38: 662-673.
[21] S. Lu, D. Yang and G. Hu. Herz Type Spaces and Their Applications[M]. Beijing:Science Press, 2008.158-168.
[22] J. Marcinkiewicz. Sur quelques inte′grales du type de Dini. Ann Soc Polon Math[J].1938, 17: 42-50.
[23] Soria F, Weiss G. A remark on singular integrals and power weights. Indiana UnivMath[J]. 1994, 43: 187-204.
[24] S. Lu and D. Yang. The decomposition of weighted Herz space on Rn and its appli-cations. Science in China (Ser. A)[J]. 1995, 38(2): 147-158.
[25] Y. Sun and W. Su, An endpoint estimate for the commutator of singular integrals.Acta Math[J]. 2005, 21(6): 1249-1258.
[26] C. Chanillo and M. Christ. Weak (1,1) bounds for oscillatory singular integrals. DukeMath J[J]. 1987, 55(1): 141-155.
[27] Y. Pan. Boundedness of oscillatory singular integrals on Hardy spaces: II. Indiana UMath J[J]. 1992, 41: 279-293.
[28] Y. Hu and Y. Pan. Boundedness of oscillatory singular integrals on Hardy spaces.Ark Math[J]. 1992, 30: 311-320.
[29] D. Phong and E. Stein. Hilbert integrals, singular integrals, and Radon transformsI. Acta Math[J]. 1986, 157(12): 99-157.
[30] G. Hu and D.Yan. On the commutator of the Marcinkiewciz integral. J Math AnalAppl[J]. 2003, 283: 351-361.
[31] L. Tang. Boundedness for the commutator of convolution operator. J Math AnalAppl[J]. 2004, 279: 545-555.
[32] Y. Ding, S. Lu and K. Yabuta. On commutators of Marcinkiewciz integrals withrough kernel. Math Anal Appl[J]. 2002, 275 (1): 66-69.
[33] D. Chen and J. Chen, The boundedness of Marcinkiewicz integral commutators withrough kernels on homogeneous Herz spaces. Acta Math Scientia[J]. 2006, 26(A): 832-839.
[34] S. Lu. Real variable theory of Hp spaces and its applications[M]. Shanghai: ShanghaiScientific & Technical Publishers, 1992.19-50.
[35] Y. Ding, S. Lu and K. Yabuta. On commutators of Marcinkiewicz integrals withrough kernel. Math Appl[J]. 2000, (1): 66-69.
[36] S. Lu, L. Tang and D. Yang. Boundedness of commutator on homogeneous Herzspaces. Science in China (Ser A)[J]. 1998, 41(10):1023-1033.
[37] B. Muckenhoupt and R. L. Wheeden. Weighted norm inequalities for singular andfractional integrals. Trans Amer Math Soc J[J]. 1971,161: 249-261.
[2] A. Benedek, A. Calder′on and R. Panzone. Convolution operators on Banach spacevalued functions. Proc Nat Acad Sci U.S.A.[J]. 1962, 48(3): 356-365.
[3] J.Chen, D. Fan and Y. Pan. A note on a Marcinkiewicz integral operator. MathNach[J]. 2001, 227(1): 33-42.
[4] Y. Ding, D. Fan and Y. Pan. Weighted boundedness for a class of rough Marcinkiewiczintegrals. Indiana Univ Math J[J]. 1999, 48(3): 1037-1055.
[5] Y. Ding, D. Fan and Y. Pan. LP-boundedness of Marcinkiewicz integrals withHardy space function kernels. Acta Math Sinica (English series)[J]. 2000, 16(4): 593-600.
[6] J. Chen, D. Fan and Y. Ying. The method of rotation and Marcinkiewicz integralson product domains. Studia Math[J]. 2002, 153(1): 41-58.
[7] J. Chen, D. Fan and Y. Ying. Rough Marcinkiewicz integrals withL(Log+L)2 kernelson product spaces. Adv Math(China)[J]. 2001, 30(2): 179-182.
[8] J. Chen, Y. Ding and D. Fan. Certain square functions on product spaces. MathNach[J]. 2001, 230(1): 5-18.
[9] Y. Ding, D. Fan and Y. Pan. Marcinkiewicz integrals with rough kernels on productspaces. Yokohama Math J[J]. 2001, 49: 1-15.
[10] A. Torchinsky and S. Wang. A note on Marcinkiewicz integral. Colloq Math[J]. 1990,60(2): 235-243.
[11] Y. Ding, S. Lu and K. Yabuta. On commutators of Marcinkiewicz integrals withrough kernel. Math Anal Appd[J]. 2002, 140: 60-68.
[12] Y. Han. Some properties of S-function and Marcinkiewicz integrals. Acta Sci NaturUniv pekingniss[J]. 1978, 5: 21-34.
[13] S. Qiu. Boundedness of Littlewood-paly operators and Marcinkiewicz integral onεα,p.Math Res Exposition[J]. 1992, 12: 41-50.
[14] Y. Wang. Littlewood-paly g- operator on weighted Campanato spaces. Journal ofHenan Normal University (Natural Seience)[J]. 1995, 23(3): 22-25.
[15] Y. Ding and S. Lu. Weighted norm inequalities for fractional integral operator withrough kernel. Can J Math[J]. 1998, 50(1): 29-39.
[16] Chanillo S.,Waton D. and Wheeden R.l. Some integral and maximal operator ralatedto starlike sets. Studia Math[J]. 1993, 107(1): 223-255.
[17] Y. Ding, S. Lu and Q. Xue. On Marcinkiewicz integral with homogeneous kernels. JMath Anal and Appl[J]. 2000, 245(2): 471-488.
[18] C. Perez. Endpoint estimates for commutator of singular integral operators. J FunctAnal[J]. 1995, 128: 163-185.
[19] S. Lu, and D. Yang. The local version of Hp spaces at the origin. studia Math[J].1995, 116: 103-131.
[20] S. Lu and D. Yang. The weighted Herz-type Hardy spaces and its applications. Sciencein China(Ser. A)(English)[J]. 1995, 38: 662-673.
[21] S. Lu, D. Yang and G. Hu. Herz Type Spaces and Their Applications[M]. Beijing:Science Press, 2008.158-168.
[22] J. Marcinkiewicz. Sur quelques inte′grales du type de Dini. Ann Soc Polon Math[J].1938, 17: 42-50.
[23] Soria F, Weiss G. A remark on singular integrals and power weights. Indiana UnivMath[J]. 1994, 43: 187-204.
[24] S. Lu and D. Yang. The decomposition of weighted Herz space on Rn and its appli-cations. Science in China (Ser. A)[J]. 1995, 38(2): 147-158.
[25] Y. Sun and W. Su, An endpoint estimate for the commutator of singular integrals.Acta Math[J]. 2005, 21(6): 1249-1258.
[26] C. Chanillo and M. Christ. Weak (1,1) bounds for oscillatory singular integrals. DukeMath J[J]. 1987, 55(1): 141-155.
[27] Y. Pan. Boundedness of oscillatory singular integrals on Hardy spaces: II. Indiana UMath J[J]. 1992, 41: 279-293.
[28] Y. Hu and Y. Pan. Boundedness of oscillatory singular integrals on Hardy spaces.Ark Math[J]. 1992, 30: 311-320.
[29] D. Phong and E. Stein. Hilbert integrals, singular integrals, and Radon transformsI. Acta Math[J]. 1986, 157(12): 99-157.
[30] G. Hu and D.Yan. On the commutator of the Marcinkiewciz integral. J Math AnalAppl[J]. 2003, 283: 351-361.
[31] L. Tang. Boundedness for the commutator of convolution operator. J Math AnalAppl[J]. 2004, 279: 545-555.
[32] Y. Ding, S. Lu and K. Yabuta. On commutators of Marcinkiewciz integrals withrough kernel. Math Anal Appl[J]. 2002, 275 (1): 66-69.
[33] D. Chen and J. Chen, The boundedness of Marcinkiewicz integral commutators withrough kernels on homogeneous Herz spaces. Acta Math Scientia[J]. 2006, 26(A): 832-839.
[34] S. Lu. Real variable theory of Hp spaces and its applications[M]. Shanghai: ShanghaiScientific & Technical Publishers, 1992.19-50.
[35] Y. Ding, S. Lu and K. Yabuta. On commutators of Marcinkiewicz integrals withrough kernel. Math Appl[J]. 2000, (1): 66-69.
[36] S. Lu, L. Tang and D. Yang. Boundedness of commutator on homogeneous Herzspaces. Science in China (Ser A)[J]. 1998, 41(10):1023-1033.
[37] B. Muckenhoupt and R. L. Wheeden. Weighted norm inequalities for singular andfractional integrals. Trans Amer Math Soc J[J]. 1971,161: 249-261.