空间的刻画与奇异积分算子的权模不等式
|
摘要
H~p空间的实变理论是上世纪70年代以来调和分析中最富有成果的领域之一。该理论运用同复变或调和函数方法无关的多种形式的极大函数来刻画H~p空间的特征。这个理论的深入发展阶段便是H~p空间的分解结构理论的建立。分解结构理论的思想是从微观的观点来看待函数空间，也就是把H~p空间的元素看成是一列“基本元素”依某种形式的重叠。根据分解结构理论，人们可以将调和分析中的许多问题归结于很简单的情形。
许多空间的分解结构理论已经相当完善了，加权空间的部分分解理论也已完成。加权Herz型Hardy空间的原子刻画已在1995年由陆善镇和杨大春给出，但迄今为止未见加权Herz型Hardy空间的分子刻画。本文首先解决了加权Herz型Hardy空间的分子刻画，作为应用，给出了强奇异积分算子T_b在加权Herz型Hardy空间上的有界性的证明。
随着空间分解结构理论的日臻完善，奇异积分算子的有界性的研究也取得了空前丰硕的成果。但对于Yabuta在[37]中引进的具有深刻的微分方程背景的θ(t)型奇异积分算子的研究相对少一些，原因之一是它相对复杂一些，对于加权的情形有些有一定的难度，对于其在Banach值空间上的有界性问题讨论更少，原因之一是Banach值空间不完全具备通常的实空间的好性质。本文借助于Calderón-Zygmund分解理论和Hardy空间的分解理论，经过精细的讨论，得到了θ(t)型奇异积分算子在Banach值加权空间L_(B，w)~p(R~n)(1≤p＜∞)上的有界性，以及在Banach值加权空间H_(B，w)~1(R~n)上的有界性。
最后，本文运用加权Hardy空间的分子刻画理论，讨论了θ(t)型Calderón-Zygmund算子在加权Hardy空间上的有界性。证明了θ(t)型Calderón-Zygmund算子是从H_w~p到H_w~p有界的，以及从H_w~1到L_w~1有界的。
The real variable theory of the Hp space is one of the most wealthiest fields of harmonic analysis from 1970's. The theory apply many kinds of maximal functions to characterize Hp space, this method is different from the complex method or the hamonic function method. The deeply developing stage about this theory is establishing the characterization structural theory of Hp space. The idea of the characterization structural theory is looking at the function spaces from microcosmic viewpoint, in other words regarding the element of Hp space as a overlap of a series of "basic element" According as the characterization structural theory we can make many problems about the hamonic analysis coming down to the simple instances.
Many spaces's characterization structural theory have developed quite perfect, and part of the weighted spaces's characterization structural theory have fulfilled. In 1995, Lu Shanzhen and Yang Dachun had given the atom characterization of the weighted Herz type Hardy space, but so far there is not the molecole characterization of the weighted Herz type Hardy space. At first this paper solved the molecole characterization of the weighted Herz type Hardy space, as an application, author also gave the proof about the strong singular integral operator Tb's boundedness in weighted Herz type Hardy space.
Along with the developing of the molecole characterization of the weighted Herz type Hardy space, the study of the singular integral operator get a plentiful harvest. But there is only a few fruits about the (t) type singular integral operator which had been braught forward by Yabuta in [37], and this operator has a profound partial differential coefficient background. The reason is that it is relatively complex. For the weighted case some operators have a little difficulty; For the case of the boundedness of this operator in Banach-value space, there is less study because the Banach-value space don't possess the good properties of the normal spaces. At last this paper recur to the Calderon-Zygmund characterization theory and the characterization theory of Hardy space, utilize the fine discussion, we get the 6(t) type singular integral operator's boundedness in the Banach-value weighted space LPB, (Rn) (1 p < ) and the Banach-value weighted space H1B, (Rn).
At last, this paper use the molecole characterization of the weighted Hardy space to discuss the (t) type Calderon-Zygmund operator's boundedness in the weighted Hardy space, and proveing that (t) type Calder n-Zygmund operator is boundedness from Hp to Hp and from H1 to L1.
许多空间的分解结构理论已经相当完善了，加权空间的部分分解理论也已完成。加权Herz型Hardy空间的原子刻画已在1995年由陆善镇和杨大春给出，但迄今为止未见加权Herz型Hardy空间的分子刻画。本文首先解决了加权Herz型Hardy空间的分子刻画，作为应用，给出了强奇异积分算子T_b在加权Herz型Hardy空间上的有界性的证明。
随着空间分解结构理论的日臻完善，奇异积分算子的有界性的研究也取得了空前丰硕的成果。但对于Yabuta在[37]中引进的具有深刻的微分方程背景的θ(t)型奇异积分算子的研究相对少一些，原因之一是它相对复杂一些，对于加权的情形有些有一定的难度，对于其在Banach值空间上的有界性问题讨论更少，原因之一是Banach值空间不完全具备通常的实空间的好性质。本文借助于Calderón-Zygmund分解理论和Hardy空间的分解理论，经过精细的讨论，得到了θ(t)型奇异积分算子在Banach值加权空间L_(B，w)~p(R~n)(1≤p＜∞)上的有界性，以及在Banach值加权空间H_(B，w)~1(R~n)上的有界性。
最后，本文运用加权Hardy空间的分子刻画理论，讨论了θ(t)型Calderón-Zygmund算子在加权Hardy空间上的有界性。证明了θ(t)型Calderón-Zygmund算子是从H_w~p到H_w~p有界的，以及从H_w~1到L_w~1有界的。
The real variable theory of the Hp space is one of the most wealthiest fields of harmonic analysis from 1970's. The theory apply many kinds of maximal functions to characterize Hp space, this method is different from the complex method or the hamonic function method. The deeply developing stage about this theory is establishing the characterization structural theory of Hp space. The idea of the characterization structural theory is looking at the function spaces from microcosmic viewpoint, in other words regarding the element of Hp space as a overlap of a series of "basic element" According as the characterization structural theory we can make many problems about the hamonic analysis coming down to the simple instances.
Many spaces's characterization structural theory have developed quite perfect, and part of the weighted spaces's characterization structural theory have fulfilled. In 1995, Lu Shanzhen and Yang Dachun had given the atom characterization of the weighted Herz type Hardy space, but so far there is not the molecole characterization of the weighted Herz type Hardy space. At first this paper solved the molecole characterization of the weighted Herz type Hardy space, as an application, author also gave the proof about the strong singular integral operator Tb's boundedness in weighted Herz type Hardy space.
Along with the developing of the molecole characterization of the weighted Herz type Hardy space, the study of the singular integral operator get a plentiful harvest. But there is only a few fruits about the (t) type singular integral operator which had been braught forward by Yabuta in [37], and this operator has a profound partial differential coefficient background. The reason is that it is relatively complex. For the weighted case some operators have a little difficulty; For the case of the boundedness of this operator in Banach-value space, there is less study because the Banach-value space don't possess the good properties of the normal spaces. At last this paper recur to the Calderon-Zygmund characterization theory and the characterization theory of Hardy space, utilize the fine discussion, we get the 6(t) type singular integral operator's boundedness in the Banach-value weighted space LPB, (Rn) (1 p < ) and the Banach-value weighted space H1B, (Rn).
At last, this paper use the molecole characterization of the weighted Hardy space to discuss the (t) type Calderon-Zygmund operator's boundedness in the weighted Hardy space, and proveing that (t) type Calder n-Zygmund operator is boundedness from Hp to Hp and from H1 to L1.
引文
[1] Bourgain J. Extension of a Result of Benedek, Calderón and Panzone[J] , ibid. 1984, 22(1): 91-95.
[2] Chen Y Z, Lau K S. Some new class of Hardy spaces. J. Funct. Anal., 1989, 84: 255-278.
[3] Ciofman R R, Fefferman C. Weighted Norm Inequalities for Maximal Function and Singular Integral[J] , Studia Math. 1974, 51: 241-250.
[4] 邓东皋，韩永生．H~p空间论[M] ．北京：北京大学出版社，1992．
[5] Garcia-Cuerva J. Hardy spaces and Beurling algebras. J. London Math. Soc., 1989, 39: 499-513.
[6] G. Bourdaud, L~p-estimates for certain non-regular pseudo-differential operators, Comm. Partial Differential Equations. 1982(7): 1023-1033.
[7] Garcia-Cuerva J, Rubio de Francia J L. Weighted Norm Inequalities and Related Topics[M] , North-Holland, Amsterdam, 1985. Integral[J] , Studia Math. 1974, 51: 241-250.
[8] Herz C. Lipshitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms[J] . J. Math. Mech., 1968, 18: 283-324.
[9] 韩永生．近代调和分析方法及其应用[M] ．北京：科学出版社，1988．
[10] José L. Rubio de Francia. Fourier Series and Hilbert Transforms with Values in UMD Banach Spaces[J] . Studia Math. 1985, 81(1): 95-105.
[11] Kuang Jichang, On Weighted Hp Boundedness of C-Z Type Singular Integral Operators[J] , Journal of Mathematical Analysis and Applications, 1996(197): 864-874.
[12] Lu S Z, Taibleson M H, Weiss G. On the almost everywhere convergence of Bochner-Riesz means of multiple Fourier series[J] . Lecture Note in Math., 1982, 908: 311-318.
[13] Lu S Z, Taibleson M H, Weiss G. Spaces Generated by Blocks[M] . Publishing House of Beijing Normal University, 1989.
[14] Lu S Z, Yang D C. The decomposition of the weighted Herz spaces and its application.
Science in China (Ser.A), 1995, 38: 147-158.
[15] Lu S Z, Yang D C. The weighted Herz-type Hardy spaces and its application. Science in China (Ser.A), 1995, 38: 662-673.
[16] 陆善镇，杨大春，加权Herz型Hardy的空间及其应用[J] ，中国科学，A辑；1995，25(3)：235-245．
[17] Lu Shanzhen, Yang Dachun, Some Characterizations of Weighted Herz-type Hardy Spaces and Their Applications, Acta Mathematica Sinica, New Series. 1997, 13(1): 45-58.
[18] Lu Shanzhen, Yang Dachun, The Local Version of H~p(R~n)Spaces at the Origin, Studia Math [J] , 1995, 116: 103-131.
[19] 李晓春，陆善镇，加权型空间上的强奇异积分算子，数学学报，1998，41(1)：7-18．
[20] 李兴民，彭立中，加权Hardy空间的分子刻画[J] ，中国科学 2000，30(7)：583-593．
[21] 李兴民，彭立中，加权的Hardy空间和面积积分的加权模不等式[J] ，数学学报．1997．40(3)：351-356．
[22] Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function[J] . Trans. Amer. Math. Soc. 1972, 165: 207
[23] Miyachi A. Remarks on Herz type Hardy spaces. To appear in Acta Math. Sinica (English Ser.).
[24] M. Nagase, The L~p-boundedness of pseudo-differential operators with smooth symbols, Comm. Partial Differential Equations. 1977(2):1045-1061.
[25] Ming-yi Lee, Chin-cheng Lin, The Molecular Characterization of Werghted Hardy Spaces[J] , Journal of Functional Analysis, 2002(188): 442-460. Applications, 1996(197): 864-874.
[26] Mitcheu H Taibleson and Guido Weiss, The Molecular Characterization of Certain Hardy Spaces [M] , 1980, 77: 68-194.
[27] Ming-Yi Lee and Chin-Cheng Lin, The Molecular Characterization of Weighted Hardy Spaces, Journal of Functional Analysis[J] , 2002, 188: 442-460.
[28] 彭立中．广义Calderóon-Zygmund算子及其加权模不等式[J] ．数学进展，1985，
14(2)：97-115．
[29] Quek Tong Seng，杨大春，R~n上加权弱Hardy空间中的Calderón-Zygmund算子[J] ，2001，44(3)：517-526．
[30] R. R. Coifman and Y. Meyer. Au delá des opérateurs pseudo-differentials, Astérisqu, 57, Soc. Math. France. 1978.
[31] Stein E M. Note on singular integrals[J] . Proc. Amer. Soc. Math., 1957, 8: 250.
[32] Soria F, Weiss G. Aremark on singular integuals and power weights[J] . Indiana Univ. Math., 1994, 43(5): 187-204.
[33] S. Mossaheb and M. Okada. Une classe a opérateurs pseudo-differentials bornés dur L′(R~n), 1＜r＜∞. C. R. Acad. Sci. Paris Sér. A 1997(285): 613-616, and Publications Mathématiques d'Orsay 77-78. 1976-1977, 41-52.
[34] Stein E. M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals[M] . Princeton: Princeton Univ Press, 1993.
[35] Taibleson M H, Weiss G. Certain function spaces associated with a.e.convergence of Fourier series[J] .Proceeding of the conference on Harmonic Analysis, in Honor of Antoni Zygmund, 1982, Ⅰ: 95-113.
[36] 韦旦．一类奇异积分算子在Banach空间值Hardy空间上的有界性[J] ．数学物理学报，1997，17(2)：235-240．
[37] Yabuta K. Generalizations of Calderón-Zygmund operators[J] .Studia math. 1985, 82(1): 17-31.
[38] Zhao Kai. On generalized Calderón-Zygmund operators[J] . Journal of Mathematical Research and Exposition, 1995, 15(2): 211-215.
[39] Zhao Kal. Some Bounded Results of θ(t)-type Singular Integral Operators[J] , App. Math. and Mech., 2000, 21(5): 573-578.
[40] 赵凯．θ(t)-Calderón-Zygmund算子在群上的空间俨中的有界性[J] ．数学研究与评论，2000，20(3)：459-463．
[41] 赵凯．向量值函数的奇异积分算子[J] ．青岛大学学报(自然)，1992，5(1)：41-49．
[42] 赵凯，广义Calderón-Zygmund算子在加权Hardy空间的有界性[J] ，应用数学，1998，11(3)：23-26．
[2] Chen Y Z, Lau K S. Some new class of Hardy spaces. J. Funct. Anal., 1989, 84: 255-278.
[3] Ciofman R R, Fefferman C. Weighted Norm Inequalities for Maximal Function and Singular Integral[J] , Studia Math. 1974, 51: 241-250.
[4] 邓东皋，韩永生．H~p空间论[M] ．北京：北京大学出版社，1992．
[5] Garcia-Cuerva J. Hardy spaces and Beurling algebras. J. London Math. Soc., 1989, 39: 499-513.
[6] G. Bourdaud, L~p-estimates for certain non-regular pseudo-differential operators, Comm. Partial Differential Equations. 1982(7): 1023-1033.
[7] Garcia-Cuerva J, Rubio de Francia J L. Weighted Norm Inequalities and Related Topics[M] , North-Holland, Amsterdam, 1985. Integral[J] , Studia Math. 1974, 51: 241-250.
[8] Herz C. Lipshitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms[J] . J. Math. Mech., 1968, 18: 283-324.
[9] 韩永生．近代调和分析方法及其应用[M] ．北京：科学出版社，1988．
[10] José L. Rubio de Francia. Fourier Series and Hilbert Transforms with Values in UMD Banach Spaces[J] . Studia Math. 1985, 81(1): 95-105.
[11] Kuang Jichang, On Weighted Hp Boundedness of C-Z Type Singular Integral Operators[J] , Journal of Mathematical Analysis and Applications, 1996(197): 864-874.
[12] Lu S Z, Taibleson M H, Weiss G. On the almost everywhere convergence of Bochner-Riesz means of multiple Fourier series[J] . Lecture Note in Math., 1982, 908: 311-318.
[13] Lu S Z, Taibleson M H, Weiss G. Spaces Generated by Blocks[M] . Publishing House of Beijing Normal University, 1989.
[14] Lu S Z, Yang D C. The decomposition of the weighted Herz spaces and its application.
Science in China (Ser.A), 1995, 38: 147-158.
[15] Lu S Z, Yang D C. The weighted Herz-type Hardy spaces and its application. Science in China (Ser.A), 1995, 38: 662-673.
[16] 陆善镇，杨大春，加权Herz型Hardy的空间及其应用[J] ，中国科学，A辑；1995，25(3)：235-245．
[17] Lu Shanzhen, Yang Dachun, Some Characterizations of Weighted Herz-type Hardy Spaces and Their Applications, Acta Mathematica Sinica, New Series. 1997, 13(1): 45-58.
[18] Lu Shanzhen, Yang Dachun, The Local Version of H~p(R~n)Spaces at the Origin, Studia Math [J] , 1995, 116: 103-131.
[19] 李晓春，陆善镇，加权型空间上的强奇异积分算子，数学学报，1998，41(1)：7-18．
[20] 李兴民，彭立中，加权Hardy空间的分子刻画[J] ，中国科学 2000，30(7)：583-593．
[21] 李兴民，彭立中，加权的Hardy空间和面积积分的加权模不等式[J] ，数学学报．1997．40(3)：351-356．
[22] Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function[J] . Trans. Amer. Math. Soc. 1972, 165: 207
[23] Miyachi A. Remarks on Herz type Hardy spaces. To appear in Acta Math. Sinica (English Ser.).
[24] M. Nagase, The L~p-boundedness of pseudo-differential operators with smooth symbols, Comm. Partial Differential Equations. 1977(2):1045-1061.
[25] Ming-yi Lee, Chin-cheng Lin, The Molecular Characterization of Werghted Hardy Spaces[J] , Journal of Functional Analysis, 2002(188): 442-460. Applications, 1996(197): 864-874.
[26] Mitcheu H Taibleson and Guido Weiss, The Molecular Characterization of Certain Hardy Spaces [M] , 1980, 77: 68-194.
[27] Ming-Yi Lee and Chin-Cheng Lin, The Molecular Characterization of Weighted Hardy Spaces, Journal of Functional Analysis[J] , 2002, 188: 442-460.
[28] 彭立中．广义Calderóon-Zygmund算子及其加权模不等式[J] ．数学进展，1985，
14(2)：97-115．
[29] Quek Tong Seng，杨大春，R~n上加权弱Hardy空间中的Calderón-Zygmund算子[J] ，2001，44(3)：517-526．
[30] R. R. Coifman and Y. Meyer. Au delá des opérateurs pseudo-differentials, Astérisqu, 57, Soc. Math. France. 1978.
[31] Stein E M. Note on singular integrals[J] . Proc. Amer. Soc. Math., 1957, 8: 250.
[32] Soria F, Weiss G. Aremark on singular integuals and power weights[J] . Indiana Univ. Math., 1994, 43(5): 187-204.
[33] S. Mossaheb and M. Okada. Une classe a opérateurs pseudo-differentials bornés dur L′(R~n), 1＜r＜∞. C. R. Acad. Sci. Paris Sér. A 1997(285): 613-616, and Publications Mathématiques d'Orsay 77-78. 1976-1977, 41-52.
[34] Stein E. M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals[M] . Princeton: Princeton Univ Press, 1993.
[35] Taibleson M H, Weiss G. Certain function spaces associated with a.e.convergence of Fourier series[J] .Proceeding of the conference on Harmonic Analysis, in Honor of Antoni Zygmund, 1982, Ⅰ: 95-113.
[36] 韦旦．一类奇异积分算子在Banach空间值Hardy空间上的有界性[J] ．数学物理学报，1997，17(2)：235-240．
[37] Yabuta K. Generalizations of Calderón-Zygmund operators[J] .Studia math. 1985, 82(1): 17-31.
[38] Zhao Kai. On generalized Calderón-Zygmund operators[J] . Journal of Mathematical Research and Exposition, 1995, 15(2): 211-215.
[39] Zhao Kal. Some Bounded Results of θ(t)-type Singular Integral Operators[J] , App. Math. and Mech., 2000, 21(5): 573-578.
[40] 赵凯．θ(t)-Calderón-Zygmund算子在群上的空间俨中的有界性[J] ．数学研究与评论，2000，20(3)：459-463．
[41] 赵凯．向量值函数的奇异积分算子[J] ．青岛大学学报(自然)，1992，5(1)：41-49．
[42] 赵凯，广义Calderón-Zygmund算子在加权Hardy空间的有界性[J] ，应用数学，1998，11(3)：23-26．
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |