Bochner-Riesz算子与Besov函数生成的交换子的性质
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摘要
本文研究由Bochner-Ricsz算子与Besov函数生成的交换子T_(λ;b)~T在某些可积函数空间L~s(R~n)(s≥2)中的几乎处处收敛性,同时讨论T_(λ;b)在L~s(R~n)和L~s(R~n)中径向函数类上的有界性问题.
我们首先研究求和指标λ小于临界阶λ_0=(n-1)/2,且max{2,(?)}≤s<2n/(n-1-2λ)时,交换子T_(λ;b)~T在L~s(R~n)上的几乎处处收敛性.为了得到这一结果,我们利用紧支光滑函数的乘子算子与Besov函数生成交换子的相应的极大算子,分别进行(2,2)的有界性估计和权函数为幂权的双权(2,2)有界性估计.
其次,我们研究了当指标0<λ<(n-1)/2时,交换子T_(λ;b)在L~s(R~n)和L~s(R~n)中径向函数类上的有界性,其中s均满足|(?)|<(?)+(?).为此,我们利用Bochner-Riesz算子的分解形式T_(λ)f(x)=(?)*f(x),由对偶性及插值定理得到了新的结论.
文中深刻阐明了Bochner-Riesz算子的求和指标λ,Besov空间中的相关指数β,p,q与可积空间的指标s和d之间的相互关系.
For some integrable functions in L~s(R~n) (s≥2), in this paper we research the almost everywhere convergence of the commutators T_(λ;b)~T. generated by Bochner-Riesz operator and Besov functions, as well as the boundedness of T_(λ;b) on L~s(R~n) and on L~s(R~n, r), where L~s(R~n, r) denote the class of radical functions in L~s(R~n).
Firstly, when the summation index A is under the critical orderλ_0= (n-1)/2, and max(2, (?))≤s < 2n/(n-1-2λ), we study the almost everywhere convergence of the commutator T_(λ;b)~T on L~s(R~n). To get this result, we research the (2, 2) boundedness and the two-weighted (2, 2) boundedness of the maximal operator of the commutator, which are generated by the multipliers of compactly supported smooth functions and Besov functions.
Secondly, when the index 0 <λ< (n-1)/2, we studies the boundedness of the commutator T_(λ;b)~T on L~s(R~n) and on L~s(R~n, r), where index s always satisfies |(?)| < (?)+(?) Therefore, we make use of the decomposition of Bochner-Riesz operator to achieve some new results directly by duality and interpolation.
The paper indicates the relationship among the summation index of the Bochner-Riesz operatorλ, the index of Besov spaceβ, p, q and the integrable spaces of order s and d profoundly.
我们首先研究求和指标λ小于临界阶λ_0=(n-1)/2,且max{2,(?)}≤s<2n/(n-1-2λ)时,交换子T_(λ;b)~T在L~s(R~n)上的几乎处处收敛性.为了得到这一结果,我们利用紧支光滑函数的乘子算子与Besov函数生成交换子的相应的极大算子,分别进行(2,2)的有界性估计和权函数为幂权的双权(2,2)有界性估计.
其次,我们研究了当指标0<λ<(n-1)/2时,交换子T_(λ;b)在L~s(R~n)和L~s(R~n)中径向函数类上的有界性,其中s均满足|(?)|<(?)+(?).为此,我们利用Bochner-Riesz算子的分解形式T_(λ)f(x)=(?)*f(x),由对偶性及插值定理得到了新的结论.
文中深刻阐明了Bochner-Riesz算子的求和指标λ,Besov空间中的相关指数β,p,q与可积空间的指标s和d之间的相互关系.
For some integrable functions in L~s(R~n) (s≥2), in this paper we research the almost everywhere convergence of the commutators T_(λ;b)~T. generated by Bochner-Riesz operator and Besov functions, as well as the boundedness of T_(λ;b) on L~s(R~n) and on L~s(R~n, r), where L~s(R~n, r) denote the class of radical functions in L~s(R~n).
Firstly, when the summation index A is under the critical orderλ_0= (n-1)/2, and max(2, (?))≤s < 2n/(n-1-2λ), we study the almost everywhere convergence of the commutator T_(λ;b)~T on L~s(R~n). To get this result, we research the (2, 2) boundedness and the two-weighted (2, 2) boundedness of the maximal operator of the commutator, which are generated by the multipliers of compactly supported smooth functions and Besov functions.
Secondly, when the index 0 <λ< (n-1)/2, we studies the boundedness of the commutator T_(λ;b)~T on L~s(R~n) and on L~s(R~n, r), where index s always satisfies |(?)| < (?)+(?) Therefore, we make use of the decomposition of Bochner-Riesz operator to achieve some new results directly by duality and interpolation.
The paper indicates the relationship among the summation index of the Bochner-Riesz operatorλ, the index of Besov spaceβ, p, q and the integrable spaces of order s and d profoundly.
引文
[1] Calderon A P. Commutators of singular integral operators. Proc. Nat. Acad. Sci. USA., 1965,53(5): 1092-1099
[2] Bajsanski B, Coifman R. On singular integrals. Amer Maht Soc Providence R I, 1967, 10:1-17
[3] Cohen J, Gosselin J. A BMO estimate for multilinear singular integrals. ILLinois J Math,1986, 30: 445-464
[4] Cohen J, Gosselin J. One multilinear singular integrals on R~n. Studia Math, 1982, 72:117-133
[5] Hofman S. On certain non-stand Calderon-Zygmund operators. Studia Math. 1994, 109:105-131
[6] Hu G E. Weighted norm inequalities for commutators of homogeneous singular integrals.Acta Math Sinica Special Issue, 1995, 11: 77-88
[7] Ma B L Hu G E. Maximal operators associated with commutators of spherical means. Tohoku Mathemetrics Jounal, 1998, 50(3): 349-363
[8] Coifman R, Rochberg R and Weiss G. Facorization theoems for Hardy spaces in scvel Variables. Ann of Math, 1976, 102: 611-635
[9] Janson S. Mean oscillation of singular integral operators. Ark Math, 1978, 16: 263-270
[10] Sampson G, Naparstek A, Drobot V. (L~s,L~d) mapping properies of convolition transforms.Studia Math. 1976, (55): 41-70
[11] Xia X. Boundedness of some singular integral operators and their commutators: Ph. D.Thesis Beijing Normal Univ. Beijing: Beijing Normal Univ., 2007, 10-28
[12] Stein E M. Singular integrals and differentiability of function. Princeton Univ Press N J.,1970
[13] Chanillo S. A note on commutators. Indiana Univ Math J., 1982, 31: 7-16
[14] Paluszynski M. Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J., 1995, 44(1): 1-17
[15] 陆善镇,吴强,杨大春.交换子在Hardy空间上的有界性.中国科学(A辑),2002,32(3):232-270
[16] Peez C, Trujillo-gonzalez R. Sharp weighted estimates for multilinear commutators, London Math Soc, 2002, 65: 672-692
[17] Chen Y, Ma B. The Boundedness of higher order commutators related to functions in Besov spaces. Beijing: Beijing Normal Univ., 2006. 42(1): 32-34
[18] Muckenhoupt B. Weighted norm inequalities for the Harday maximal function. Trans. Amer.Math. Soc, 1972, 165(1): 207-226
[19] Kurtz D S. Littlewood-Paley and multiplier theorems on weighted L~p spaces. Trans. Amer.Math. Soc, 1980, 209(1): 235-254
[20] Chanillo S. Weighted norm inequalities for strongly singular convolution operators. Trans.Amer. Math. Soc, 1984, 281(1): 77-107
[21] Bloom S. A commutator theorem and weighted BMO. Trans. Amer. Math. Soc, 1985,292(1): 103-122
[22] You Z. Results on commutators obtained from weighted norm inequalities. Chinese Adv. in Math., 1988, 17(1): 79-84
[23] Ga(?)cia-Cuerva J, Harboure E, Segovia C, Torrea J L. Weighted norm inequalities for commutators of strongly singular integrals. Indiana Univ. Math. J., 1991, 40(44): 1397-1420
[24] Alvarez J, Bagby R J, Kurtz D S, P(?)rez C. Weighted estimates for commutators of linear operators. Studia Math., 1993, 104(1): 195-209
[25] Hirschman 11. Multiplier transformations. Duke Math. Jour., 1959, 26(2): 222-242
[26] Carleson L. Sj(?)lin P, Oscillatory integrals and a multiplier problem for the disc. Studia Math., 1972, 44(3): 287-299
[27] Feffernan C. A note on the spherical summation multiplier. Israel J. Math., 1973, 15(1):44-52
[28] Carbery A. The boundedness of the maximal Bochner-Riesz operator on L~4(R~2). Duke Math. J., 1983, 50(2): 409-416
[29] Christ M. On almost-everywhere convergence of Bochner-Riesz means in higher dimensions.Proc Amer. Math. Soc, 1985, 95(1): 16-20
[30] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonolity and Oscillatory Integrals. Princeton Univ. Press, Princeton, New Jersey, 1993, 10-42
[31] Hu G, Lu S. The commutator of the Bochner-Riesz operator. T(?)hoku Math. J., 1996, 48(2):259-266
[32] Hu G, Lu S. The maximal operator associated with the commutator of the Bochner-Riesz operator. Beijing Math., 1996, 2(1): 96-106
[33] Hu G, Lu S. A weighted L~2 estimates for the commutator of the Bochner-Riesz operator.Proc Amer. Math. Soc, 1997, 125(10): 2867-2873
[34] Liu L Z, Lu S Z. Weighted weak type inequalities for maximal commutators of Bochner-Riesz operator. Hokkaedo Math., 2003, 32(1): 85-99
[35] Liu L Z. The continuity of commutators on Tricbel-Lizorkin spaces. Integr Equ. Opcr.Theory, 2004, 49(1): 65-75
[36] Ma B, Hu G. Maximal operators associated with commutators of spherical means. Tohoku Math. J., 1998, 50(3): 349-363
[37] Hu G, Lu S, Ma B. Commutator of convolution operator. Acta. Math. Sinica, 1999, 42(2):359-368
[38] Hu G, Ma B. L~2(R~n) Boundedness for the Commutators of Homogeneous Singular Integral Operators. Acta. Math. Sinica, 2004, 20(1): 785-792
[39] 曹前.粗糙核奇异积分算子及振荡积分算子生成的交换子的有界性:湖南大学硕士学位论文.湖南长沙:湖南大学,2007,1-31
[40] 高雄略.Besov函数与卷积算子交换子的有界性问题:湖南大学硕士学位论文.湖南长沙:湖南大学,2007,1-32
[41] Liu L X, Ma B L. The commutators of the multiplier operators. Pan. Amer. Math., 2008,23(2): 45-56
[42] 龚淑丽.Lipschitz函数与Bochner-Riesz算子生成的交换子的性质:湖南大学硕士论文.湖南长沙:湖南大学,2008,1-30
[43] Shi X, Sun Q. Weighted norm inequalities for Bochner-Riesz operators and singular integral operators. Proc. Amer. Math. Soc, 1992, 116(3): 665-673
[44] Liu Z. The Lipschitz estimats for the maximal commutator of the Bochner-Riesz operator.Beijing Normal Univ., 2002, 38(2): 160-164
[45] Carbery A, Rubio de Ftancia J L, Vega L. Almost everywhere summability of Fourier integrals. J. LondonMath. Soc, 1998, 38(33): 513-524
[46] Cordoba A. The Kakeya maximal function and the spherical summation multiplier. Amer.J. Math., 1977, 99(1): 1-22
[47] Stein E M. Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton univ.Press, Princeton, New Jersey, 1970, 15-25
[48] Hoffman S. Weighted norm inequalities and vector-valued inequalities for certain rough operators. Indiana Univ. Math. J., 1993, 42(1): 1-14
[49] Bergh J, L(?)fstr(?)m J. Interpolation Spaces-an Introduction. Springer-Verlag, New York,1976, 58(223): 23-49
[2] Bajsanski B, Coifman R. On singular integrals. Amer Maht Soc Providence R I, 1967, 10:1-17
[3] Cohen J, Gosselin J. A BMO estimate for multilinear singular integrals. ILLinois J Math,1986, 30: 445-464
[4] Cohen J, Gosselin J. One multilinear singular integrals on R~n. Studia Math, 1982, 72:117-133
[5] Hofman S. On certain non-stand Calderon-Zygmund operators. Studia Math. 1994, 109:105-131
[6] Hu G E. Weighted norm inequalities for commutators of homogeneous singular integrals.Acta Math Sinica Special Issue, 1995, 11: 77-88
[7] Ma B L Hu G E. Maximal operators associated with commutators of spherical means. Tohoku Mathemetrics Jounal, 1998, 50(3): 349-363
[8] Coifman R, Rochberg R and Weiss G. Facorization theoems for Hardy spaces in scvel Variables. Ann of Math, 1976, 102: 611-635
[9] Janson S. Mean oscillation of singular integral operators. Ark Math, 1978, 16: 263-270
[10] Sampson G, Naparstek A, Drobot V. (L~s,L~d) mapping properies of convolition transforms.Studia Math. 1976, (55): 41-70
[11] Xia X. Boundedness of some singular integral operators and their commutators: Ph. D.Thesis Beijing Normal Univ. Beijing: Beijing Normal Univ., 2007, 10-28
[12] Stein E M. Singular integrals and differentiability of function. Princeton Univ Press N J.,1970
[13] Chanillo S. A note on commutators. Indiana Univ Math J., 1982, 31: 7-16
[14] Paluszynski M. Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J., 1995, 44(1): 1-17
[15] 陆善镇,吴强,杨大春.交换子在Hardy空间上的有界性.中国科学(A辑),2002,32(3):232-270
[16] Peez C, Trujillo-gonzalez R. Sharp weighted estimates for multilinear commutators, London Math Soc, 2002, 65: 672-692
[17] Chen Y, Ma B. The Boundedness of higher order commutators related to functions in Besov spaces. Beijing: Beijing Normal Univ., 2006. 42(1): 32-34
[18] Muckenhoupt B. Weighted norm inequalities for the Harday maximal function. Trans. Amer.Math. Soc, 1972, 165(1): 207-226
[19] Kurtz D S. Littlewood-Paley and multiplier theorems on weighted L~p spaces. Trans. Amer.Math. Soc, 1980, 209(1): 235-254
[20] Chanillo S. Weighted norm inequalities for strongly singular convolution operators. Trans.Amer. Math. Soc, 1984, 281(1): 77-107
[21] Bloom S. A commutator theorem and weighted BMO. Trans. Amer. Math. Soc, 1985,292(1): 103-122
[22] You Z. Results on commutators obtained from weighted norm inequalities. Chinese Adv. in Math., 1988, 17(1): 79-84
[23] Ga(?)cia-Cuerva J, Harboure E, Segovia C, Torrea J L. Weighted norm inequalities for commutators of strongly singular integrals. Indiana Univ. Math. J., 1991, 40(44): 1397-1420
[24] Alvarez J, Bagby R J, Kurtz D S, P(?)rez C. Weighted estimates for commutators of linear operators. Studia Math., 1993, 104(1): 195-209
[25] Hirschman 11. Multiplier transformations. Duke Math. Jour., 1959, 26(2): 222-242
[26] Carleson L. Sj(?)lin P, Oscillatory integrals and a multiplier problem for the disc. Studia Math., 1972, 44(3): 287-299
[27] Feffernan C. A note on the spherical summation multiplier. Israel J. Math., 1973, 15(1):44-52
[28] Carbery A. The boundedness of the maximal Bochner-Riesz operator on L~4(R~2). Duke Math. J., 1983, 50(2): 409-416
[29] Christ M. On almost-everywhere convergence of Bochner-Riesz means in higher dimensions.Proc Amer. Math. Soc, 1985, 95(1): 16-20
[30] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonolity and Oscillatory Integrals. Princeton Univ. Press, Princeton, New Jersey, 1993, 10-42
[31] Hu G, Lu S. The commutator of the Bochner-Riesz operator. T(?)hoku Math. J., 1996, 48(2):259-266
[32] Hu G, Lu S. The maximal operator associated with the commutator of the Bochner-Riesz operator. Beijing Math., 1996, 2(1): 96-106
[33] Hu G, Lu S. A weighted L~2 estimates for the commutator of the Bochner-Riesz operator.Proc Amer. Math. Soc, 1997, 125(10): 2867-2873
[34] Liu L Z, Lu S Z. Weighted weak type inequalities for maximal commutators of Bochner-Riesz operator. Hokkaedo Math., 2003, 32(1): 85-99
[35] Liu L Z. The continuity of commutators on Tricbel-Lizorkin spaces. Integr Equ. Opcr.Theory, 2004, 49(1): 65-75
[36] Ma B, Hu G. Maximal operators associated with commutators of spherical means. Tohoku Math. J., 1998, 50(3): 349-363
[37] Hu G, Lu S, Ma B. Commutator of convolution operator. Acta. Math. Sinica, 1999, 42(2):359-368
[38] Hu G, Ma B. L~2(R~n) Boundedness for the Commutators of Homogeneous Singular Integral Operators. Acta. Math. Sinica, 2004, 20(1): 785-792
[39] 曹前.粗糙核奇异积分算子及振荡积分算子生成的交换子的有界性:湖南大学硕士学位论文.湖南长沙:湖南大学,2007,1-31
[40] 高雄略.Besov函数与卷积算子交换子的有界性问题:湖南大学硕士学位论文.湖南长沙:湖南大学,2007,1-32
[41] Liu L X, Ma B L. The commutators of the multiplier operators. Pan. Amer. Math., 2008,23(2): 45-56
[42] 龚淑丽.Lipschitz函数与Bochner-Riesz算子生成的交换子的性质:湖南大学硕士论文.湖南长沙:湖南大学,2008,1-30
[43] Shi X, Sun Q. Weighted norm inequalities for Bochner-Riesz operators and singular integral operators. Proc. Amer. Math. Soc, 1992, 116(3): 665-673
[44] Liu Z. The Lipschitz estimats for the maximal commutator of the Bochner-Riesz operator.Beijing Normal Univ., 2002, 38(2): 160-164
[45] Carbery A, Rubio de Ftancia J L, Vega L. Almost everywhere summability of Fourier integrals. J. LondonMath. Soc, 1998, 38(33): 513-524
[46] Cordoba A. The Kakeya maximal function and the spherical summation multiplier. Amer.J. Math., 1977, 99(1): 1-22
[47] Stein E M. Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton univ.Press, Princeton, New Jersey, 1970, 15-25
[48] Hoffman S. Weighted norm inequalities and vector-valued inequalities for certain rough operators. Indiana Univ. Math. J., 1993, 42(1): 1-14
[49] Bergh J, L(?)fstr(?)m J. Interpolation Spaces-an Introduction. Springer-Verlag, New York,1976, 58(223): 23-49