极大交换子在各向异性空间的有界性
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摘要
本文介绍了各向异性Hardy空间和有关Hardy型空间的基础知识及理论，简单阐述了这些空间的最新进展．受齐次Morrey-Herz型函数空间的启发，引进了一类各向异性的齐次Morrey-Herz型函数空间，验证了此空间上常用的基本不等式及插值定理，并用内插法证明了各向异性空间上的Hardy-Littlewood极大交换子在L~p(R~n)空间上的有界性．由L~p(R~n)空间有界性结论，并利用Hardy-Littlewood极大交换子的性质，以及一些不等式估计，得到了Hardy-Littlewood极大交换子在各向异性的齐次Morrey-Herz型空间上的有界性结果．对于分数次极大交换子，在各向异性的齐次Morrey-Herz型空间上类似的有界性问题也得到了证明．最后还证明了满足一定尺寸条件的一类线性算子的交换子在各向异性的齐次Morrey-Herz型空间上的有界性．
In this thesis, we recall some basic knowledge and theorems of anisotropic Hardy space and Herz type Hardy spaces, and describe the development of these sapces briefly. Motivated by the theory of homogeneous Morrey-Herz type spaces, a class of anisotropic homogeneous Morrey-Herz type spaces associated with a non-isotropic dilation A on R~n are introduced , some basic inequalities and interpolation theorems are established. By the interpolation theorems, the L~p boundedness of the commutator of Hardy-Littlewood maximal founction on anisotropic spaces are obtained. Using the L~p boundedness and the properties of the Hardy-Littlewood maximal fountions, the boundedness of the commutator of Hardy-Littlewood maximal founction on the homogeneous anisotropic Morrey-Herz type spaces can be proved. The same results are hold for the commutator of fractional maximal founction. Finally, we obtain the boundedness of the commutator of a class of linear operators which satisfied some size conditions.
In this thesis, we recall some basic knowledge and theorems of anisotropic Hardy space and Herz type Hardy spaces, and describe the development of these sapces briefly. Motivated by the theory of homogeneous Morrey-Herz type spaces, a class of anisotropic homogeneous Morrey-Herz type spaces associated with a non-isotropic dilation A on R~n are introduced , some basic inequalities and interpolation theorems are established. By the interpolation theorems, the L~p boundedness of the commutator of Hardy-Littlewood maximal founction on anisotropic spaces are obtained. Using the L~p boundedness and the properties of the Hardy-Littlewood maximal fountions, the boundedness of the commutator of Hardy-Littlewood maximal founction on the homogeneous anisotropic Morrey-Herz type spaces can be proved. The same results are hold for the commutator of fractional maximal founction. Finally, we obtain the boundedness of the commutator of a class of linear operators which satisfied some size conditions.
引文
[1] Calderon A P, Torchinsky A. Parabolic maximal functions associated with a distribution[J]. Adv.in Math. 1975,16: 1-64.
[2] Calderon A P, Torchinsky A. Parabolic maximal functions associated with a distribution //[J]. Adv.in Math. 1977,24: 101-171.
[3] Folland G B, Stein E M. Hardy Spaces on Homogeneous Groups[M]. Princeton: Princeton University Press, 1982.
[4] Besov O V, Il'in V P, and Nikol'skil S M. Integral Representations of Functions and Imbedding Theorems[M]. vol.Ⅰ and Ⅱ, Washington, V.H.Winston & Sons, 1979.
[5] Schmeisser H J, Triebel H. Topics in Fourier Analysis and Function Spaces[M]. Chichester,John Wiley & Sons, 1987.
[6] Triebel H. Theory of Function Spaces[M]. Monographs in Math, no.78, Birkhauser Verlag, Basel, 1983.
[7] Triebel H. Theory of Function Spaces JJ[M]. Monographs in Math, no. 84, Birkhauser Verlag, Basel, 1992.
[8] Triebel H. Wavelet bases in anisotropic function spaces [J]. Function Spaces, Differential Operators and Nonlinear Analysis, 2004: 370-387.
[9] Farkas W. Atomic and subatomic decompositions in anisotropic function spaces[J]. Math. Nachr. 2000,209: 81-113.
[10] Bownik M. Anisotropic Hardy spaces and wavelets [J]. Mem. Amer. Math. Soc. 2003,164, no. 781, 122pp.
[11] Fefferman C, Stein E M. H~p spaces of several variables [J]. Acta Math. 1972,129: 137-193.
[12]蓝森华．各向异性Hardy型空间[D]．北京师范大学， 2007年4月．
[13] Lu Shanzhen,Yang Dachun. The continuity of commutators On Herz type spaces[J]. Michigan Math J, 1997,44: 255-281.
[14] Liu Zongguang. Boundedness of commutators of Calderon-Zygmund singular integral operators On Herz-Type Hardy spaces[J]. Advances in Math, 2001,30(5): 447-458.
[15] Yabuta K. Generalizations of Calderon-Zygmund operators[J]. Studia mathematica, 1985,82: 17-31.
[16]彭立中．广义Calder(?)n—Zygmund算子及其加权模不等式[J]．数学进展， 1985，14(2)：97-115．
[17]赵凯，马丽敏，周淑娟．伴随BMO函数的交换子在Herz型Hardy空间 上的加权有界性[J]．应用数学，2004，17(3)：424-431．
[18] Morrey C B Jr. On the solutions of quasi-linear partial differential equations [J]. Trans Amer Math Soc, 1938,43: 66-126.
[19] Fazio G Di, Ragusa M A. Interior estimates in Morrey spaces for strong solutions to nondivergence from equations with discontinuous coefficientsm [J]. J Funct Anal, 1993: 112-241.
[20] Mizuhara T. Boundedness of some classical operators on generalized Morrey spaces in harmonic analysis [M]. Tokyo:Springer-Verlag, 1991: 183.
[21] Lu Shanzhen,Yang Dachun,Zhou Zusheng. Sublinear operators with rough kernel on generalized Morrey spaces[J]. Hokkaido Math J, 1998: 27-219.
[22] Hardy G H. The mean value of the modulus of an analytic function[J]. Proc.London Math.Soc. 1914,14: 269-277.
[23] Stein E M, Weiss G. On the theory of harmonic functions of several variables , I: The theory of H~p spaces[J]. Acta Math. 1960,103: 25-62.
[24] Coifman R R. A real variable characterization of H~p[J]. Studia Math. 1974,51: 269-274.
[25] Latter R H. A characterization of H~p(R~n) in terms of atoms[J]. Studia Math. 1978,62: 93-101.
[26] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality , and Oscillatory Integrals[M]. Princeton Mathematical Series,no.43, Princeton, Princeton University Press, 1993.
[27] Bownik M. Atomic and molecular decompositions of anisotropic Besov spaces[J]. Math.Zeit. 2005,250: 539-571.
[28] Bownik M. Duality and interpolation of anisotropic Triebel-Lizorkin spaces. preprint, 2006.
[29] Bownik M. Anisotropic Triebel-Lizorkin spaces with doubling measures. preprint, 2006.
[30] Bownik M, Ho K P. Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces[J]. Trans.Amer.Math.Soc. 2006,358: 1469-1510.
[31] Ho K P. Anisotropic Function Spaces[M]. Ph.D.Dissertation,Washington Univesity, 2002.
[32] Baernstein A, Sawyer E T. Embedding and multiplier theorems for H~p(R~n)[J]. Mem.Amer.Math.Soc. 1985,53,no.318,82pp.
[33] Beurling A. Construction and analysis of some convolution algebras [J]. Ann.Inst.Fourier Grenoble. 1964,14: 1-32.
[34] Lu Shanzhen. Herz type spaces[J]. Adv.in Math.(China) 2004,33: 257-272.
[35] Chanillo S. A note on commutators [J]. Indiana Univ.Math.J. 1982,31: 7-16.
[36] Coifman R R, Rochberg R, and Weiss G. Factorization theorems for Hardy spaces in sevral variables[J]. Ann.of Math. 1976,103: 611-635.
[37] Bergh J, Lofstrom J. Interpolation Space: An Introduction[M]. Beijing : World Publishing Corporation, 2003: 1-130 .
[38] Lu Shan-zhen, Yang Da-chun. The weighted Herz-type Hardy space and its applications [J]. Science in China(ser.A), 1995,38(6): 662-673.
[39] Lu Shan-zhen, Tang Lin, and Yang Da-chun. Boundedness of commutators on homogeneous Herz spaces[J]. Science in China(ser.A), 1998,41(10): 1023-1033.
[40] Lu Shan-zhen, Xu Li-fang. Boundedness of rough singular integral operators on the homogenerous Morrey- Herz spaces [J]. Hokkaido Mathematical Journal, 2005, 34(2): 299-314.
[41]徐莉芳．齐次Morrey-Herz空间上交换子的有界性[J]．北京师范大学学 报：自然科学版，2004,40(3)：297-303．
[42]陶双平，武江龙，孙小春．齐次Morrey-Herz空间上的一些基本不等式和 插值定理[J]．西北师范大学学报：自然科学版， 2006,42(4)：5-10．
[43]Stein E M，Weiss G．欧氏空间上的Fourier分析引论[M]．张阳春，译． 上海：上海科学技术出版社， 1987：191-196．
[44] Carcía-Cuerva J, Harbour E, Segovia C. etal. Weighted norm inequalities for commutors of strongly singular integrals [J]. Indiana Univ Math J, 1991,40: 1397-1420.
[45]赵向青，江寅生，曹勇辉．Herz-morrey空间上的交换子[J]．兰州大学学 报：自然科学版，2006,42(5)：116-123．
[2] Calderon A P, Torchinsky A. Parabolic maximal functions associated with a distribution //[J]. Adv.in Math. 1977,24: 101-171.
[3] Folland G B, Stein E M. Hardy Spaces on Homogeneous Groups[M]. Princeton: Princeton University Press, 1982.
[4] Besov O V, Il'in V P, and Nikol'skil S M. Integral Representations of Functions and Imbedding Theorems[M]. vol.Ⅰ and Ⅱ, Washington, V.H.Winston & Sons, 1979.
[5] Schmeisser H J, Triebel H. Topics in Fourier Analysis and Function Spaces[M]. Chichester,John Wiley & Sons, 1987.
[6] Triebel H. Theory of Function Spaces[M]. Monographs in Math, no.78, Birkhauser Verlag, Basel, 1983.
[7] Triebel H. Theory of Function Spaces JJ[M]. Monographs in Math, no. 84, Birkhauser Verlag, Basel, 1992.
[8] Triebel H. Wavelet bases in anisotropic function spaces [J]. Function Spaces, Differential Operators and Nonlinear Analysis, 2004: 370-387.
[9] Farkas W. Atomic and subatomic decompositions in anisotropic function spaces[J]. Math. Nachr. 2000,209: 81-113.
[10] Bownik M. Anisotropic Hardy spaces and wavelets [J]. Mem. Amer. Math. Soc. 2003,164, no. 781, 122pp.
[11] Fefferman C, Stein E M. H~p spaces of several variables [J]. Acta Math. 1972,129: 137-193.
[12]蓝森华．各向异性Hardy型空间[D]．北京师范大学， 2007年4月．
[13] Lu Shanzhen,Yang Dachun. The continuity of commutators On Herz type spaces[J]. Michigan Math J, 1997,44: 255-281.
[14] Liu Zongguang. Boundedness of commutators of Calderon-Zygmund singular integral operators On Herz-Type Hardy spaces[J]. Advances in Math, 2001,30(5): 447-458.
[15] Yabuta K. Generalizations of Calderon-Zygmund operators[J]. Studia mathematica, 1985,82: 17-31.
[16]彭立中．广义Calder(?)n—Zygmund算子及其加权模不等式[J]．数学进展， 1985，14(2)：97-115．
[17]赵凯，马丽敏，周淑娟．伴随BMO函数的交换子在Herz型Hardy空间 上的加权有界性[J]．应用数学，2004，17(3)：424-431．
[18] Morrey C B Jr. On the solutions of quasi-linear partial differential equations [J]. Trans Amer Math Soc, 1938,43: 66-126.
[19] Fazio G Di, Ragusa M A. Interior estimates in Morrey spaces for strong solutions to nondivergence from equations with discontinuous coefficientsm [J]. J Funct Anal, 1993: 112-241.
[20] Mizuhara T. Boundedness of some classical operators on generalized Morrey spaces in harmonic analysis [M]. Tokyo:Springer-Verlag, 1991: 183.
[21] Lu Shanzhen,Yang Dachun,Zhou Zusheng. Sublinear operators with rough kernel on generalized Morrey spaces[J]. Hokkaido Math J, 1998: 27-219.
[22] Hardy G H. The mean value of the modulus of an analytic function[J]. Proc.London Math.Soc. 1914,14: 269-277.
[23] Stein E M, Weiss G. On the theory of harmonic functions of several variables , I: The theory of H~p spaces[J]. Acta Math. 1960,103: 25-62.
[24] Coifman R R. A real variable characterization of H~p[J]. Studia Math. 1974,51: 269-274.
[25] Latter R H. A characterization of H~p(R~n) in terms of atoms[J]. Studia Math. 1978,62: 93-101.
[26] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality , and Oscillatory Integrals[M]. Princeton Mathematical Series,no.43, Princeton, Princeton University Press, 1993.
[27] Bownik M. Atomic and molecular decompositions of anisotropic Besov spaces[J]. Math.Zeit. 2005,250: 539-571.
[28] Bownik M. Duality and interpolation of anisotropic Triebel-Lizorkin spaces. preprint, 2006.
[29] Bownik M. Anisotropic Triebel-Lizorkin spaces with doubling measures. preprint, 2006.
[30] Bownik M, Ho K P. Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces[J]. Trans.Amer.Math.Soc. 2006,358: 1469-1510.
[31] Ho K P. Anisotropic Function Spaces[M]. Ph.D.Dissertation,Washington Univesity, 2002.
[32] Baernstein A, Sawyer E T. Embedding and multiplier theorems for H~p(R~n)[J]. Mem.Amer.Math.Soc. 1985,53,no.318,82pp.
[33] Beurling A. Construction and analysis of some convolution algebras [J]. Ann.Inst.Fourier Grenoble. 1964,14: 1-32.
[34] Lu Shanzhen. Herz type spaces[J]. Adv.in Math.(China) 2004,33: 257-272.
[35] Chanillo S. A note on commutators [J]. Indiana Univ.Math.J. 1982,31: 7-16.
[36] Coifman R R, Rochberg R, and Weiss G. Factorization theorems for Hardy spaces in sevral variables[J]. Ann.of Math. 1976,103: 611-635.
[37] Bergh J, Lofstrom J. Interpolation Space: An Introduction[M]. Beijing : World Publishing Corporation, 2003: 1-130 .
[38] Lu Shan-zhen, Yang Da-chun. The weighted Herz-type Hardy space and its applications [J]. Science in China(ser.A), 1995,38(6): 662-673.
[39] Lu Shan-zhen, Tang Lin, and Yang Da-chun. Boundedness of commutators on homogeneous Herz spaces[J]. Science in China(ser.A), 1998,41(10): 1023-1033.
[40] Lu Shan-zhen, Xu Li-fang. Boundedness of rough singular integral operators on the homogenerous Morrey- Herz spaces [J]. Hokkaido Mathematical Journal, 2005, 34(2): 299-314.
[41]徐莉芳．齐次Morrey-Herz空间上交换子的有界性[J]．北京师范大学学 报：自然科学版，2004,40(3)：297-303．
[42]陶双平，武江龙，孙小春．齐次Morrey-Herz空间上的一些基本不等式和 插值定理[J]．西北师范大学学报：自然科学版， 2006,42(4)：5-10．
[43]Stein E M，Weiss G．欧氏空间上的Fourier分析引论[M]．张阳春，译． 上海：上海科学技术出版社， 1987：191-196．
[44] Carcía-Cuerva J, Harbour E, Segovia C. etal. Weighted norm inequalities for commutors of strongly singular integrals [J]. Indiana Univ Math J, 1991,40: 1397-1420.
[45]赵向青，江寅生，曹勇辉．Herz-morrey空间上的交换子[J]．兰州大学学 报：自然科学版，2006,42(5)：116-123．