Lipschitz函数与Bochner-Riesz算子生成的交换子的性质
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摘要
本文研究由Lipschitz函数与Bochner-Riesz平均生成的交换子T_(λ;b)~r的几乎处处收敛性,以及对某些指数p, T_(λ;b)~r从Lp(R~n)到Lq(R~n)的有界性问题.
我们首先研究当求和指标λ小于临界阶(n - 1)/2且2≤p < 2n/(n - 1 - 2λ)时,交换子T_(λ;b)~r在Lp上的几乎处处收敛性.为了得到这一结果,我们研究紧支光滑函数的乘子算子与Lipschitz函数生成的交换子的对应极大算子的(2, 2)有界性估计以及权函数为幂权的双权(2, 2)有界性估计.
其次,本文研究了当指标λ> 0时,交换子T_(λ;b)~r从Lp(R~n)到Lq(R~n)的有界性,其中2n/(n + 2β)≤p≤2n/(n ? 2β)且1/q = 1/p -β/n.为此,我们利用紧支乘子的支集测度的方法估计交换子T_(λ;b)~r从(R~n)到L2(R~n)的有界性,再利用插值定理,得到了一些新的结论.
本文深刻揭示了Bochner-Riesz平均的求和指标、Lipschitz指数与p次可积空间的指标p三者之间相互依赖的关系,得到了一些有价值的结果.
This paper researches the almost everywhere convergence of the commutator T_(λ;b)~rgenerated by Lipschitz functions and Bochner-Riesz operator and the boundedness ofT_(λ;b)~r from Lp(R~n) to Lq(R~n) for some index p.
First, we study the almost everywhere convergence of the commutator T_(λ;b)~r, whenthe sum indexλis under the critical order (n - 1)/2 and f∈Lp, 2n/(n - 1 - 2λ). Inorder to get this result, we research the (2, 2) boundedness and the two-weighted (2, 2)boundedness of the maximal operator of the commutator generated by the multipliers ofcompact smooth functions and Lipschitz functions.
Next, the paper studies the boundedness from L~p(R~n) to L~q(R~n) of the commutatorT_(λ;b)~r, when the indexλ> 0, where 2n/(n+2β)≤p≤2 and 1/q = 1/p-β/n. In order todo this, by the support measure of the compact multiplier, we get the boundedness from(?)(R~n) to L~2(R~n), then by the interpolation theorem, we finally get some new results.
The paper indicates the relationship of the sum index of the Bochner-Riesz means,Lipschitz index and the index p of the integral spaces of order p profoundly and we getsome valuable results.
我们首先研究当求和指标λ小于临界阶(n - 1)/2且2≤p < 2n/(n - 1 - 2λ)时,交换子T_(λ;b)~r在Lp上的几乎处处收敛性.为了得到这一结果,我们研究紧支光滑函数的乘子算子与Lipschitz函数生成的交换子的对应极大算子的(2, 2)有界性估计以及权函数为幂权的双权(2, 2)有界性估计.
其次,本文研究了当指标λ> 0时,交换子T_(λ;b)~r从Lp(R~n)到Lq(R~n)的有界性,其中2n/(n + 2β)≤p≤2n/(n ? 2β)且1/q = 1/p -β/n.为此,我们利用紧支乘子的支集测度的方法估计交换子T_(λ;b)~r从(R~n)到L2(R~n)的有界性,再利用插值定理,得到了一些新的结论.
本文深刻揭示了Bochner-Riesz平均的求和指标、Lipschitz指数与p次可积空间的指标p三者之间相互依赖的关系,得到了一些有价值的结果.
This paper researches the almost everywhere convergence of the commutator T_(λ;b)~rgenerated by Lipschitz functions and Bochner-Riesz operator and the boundedness ofT_(λ;b)~r from Lp(R~n) to Lq(R~n) for some index p.
First, we study the almost everywhere convergence of the commutator T_(λ;b)~r, whenthe sum indexλis under the critical order (n - 1)/2 and f∈Lp, 2n/(n - 1 - 2λ). Inorder to get this result, we research the (2, 2) boundedness and the two-weighted (2, 2)boundedness of the maximal operator of the commutator generated by the multipliers ofcompact smooth functions and Lipschitz functions.
Next, the paper studies the boundedness from L~p(R~n) to L~q(R~n) of the commutatorT_(λ;b)~r, when the indexλ> 0, where 2n/(n+2β)≤p≤2 and 1/q = 1/p-β/n. In order todo this, by the support measure of the compact multiplier, we get the boundedness from(?)(R~n) to L~2(R~n), then by the interpolation theorem, we finally get some new results.
The paper indicates the relationship of the sum index of the Bochner-Riesz means,Lipschitz index and the index p of the integral spaces of order p profoundly and we getsome valuable results.
引文
[1] Calderon A P. Commutators of singular integral operators. Proc. Nat. Acad. Sci. USA., 1965,53(5): 1092-1099
[2] Calderon A P. Cauchy integrals on Lipschitz curves and related operators. Proc. Nat. Acad.Sci. USA., 1977, 74(4): 1324-1327
[3] John F, Nirenberg L. On functions of bounded mean oscillation. Comm. Pure Appl. Math.,1961, 14(3): 415-426
[4] Coifman R, Rochberg R, Weiss G. Facorization theorems for Harday spaces in several vari-ables. Ann. of Math., 1976, 103(3): 611-635
[5] Chanillo S. A note on commutators. Indiana Unv. Math. J., 1982, 31(1): 7-16
[6] Janson S. Mean oscillation and commutaotrs of singular integral operators. Ark. Math.,1978, 16(1): 263-270
[7] Fe?erman C. Stein E M. Hp spaces of several variables. Acta. Math., 1972, 129(1): 137-193
[8] Janson S. Mean oscillation and of singular integral operators. Ark. Math. J., 1982, 31(1):7-16
[9] Perez C. Endpoint estimates for commutators of singular integral operators. J. Funct. Anal.,1995, 128(1): 163-185
[10] Muckenhoupt B. Weighted norm inequalities for the Harday maximal function. Trans. Amer.Math. Soc., 1972, 165(1): 207-226
[11] Kurtz D S. Littlewood-Paley and multiplier theorems on weighted Lp spaces. Trans. Amer.Math. Soc., 1980, 209(1): 235-254
[12] Chanillo S. Weighted norm inequalities for strongly singular convolution operators. Trans.Amer. Math. Soc., 1984, 281(1): 77-107
[13] Bloom S. A commutator theorem and weighted BMO. Trans. Amer. Math. Soc., 1985,292(1): 103-122
[14] You Z. Results on commutators obtained from weighted norm inequalities. Chinese Adv. inMath., 1988, 17(1): 79-84
[15] Ga′rcia-Cuerva J, Harboure E, Segovia C, Torrea J L. Weighted norm inequalities for com-mutators of strongly singular integrals. Indiana Univ. Math. J., 1991, 40(44):1397-1420
[16] Alvarez J, Bagby R J, Kurtz D S, P′erez C. Weighted estimates for commutators of linearoperators. Studia Math., 1993, 104(1): 195-209
[17] Rubio de Francia J L. Maximal functions and Fourier transforms. Duke Math J., 1986,53(2): 395-405
[18] Duoandikoetxea J, Rubio de Francia J L. Maximal and singular integral operators viaFourier transform estimates. Invent Math., 1996, 84(3): 541-561
[19] Hirschman I I. Multiplier transformations. Duke Math. Jour., 1959, 26(2): 222-242
[20] Carleson L. Sj¨olin P, Oscillatory integrals and a multiplier problem for the disc. StudiaMath., 1972, 44(3): 287-299
[21] Fe?ernan C. A note on the spherical summation multiplier. Israel J. Math., 1973, 15(1):44-52
[22] Carbery A. The boundedness of the maximal Bochner-Riesz operator on L4(R2). DukeMath. J., 1983, 50(2): 409-416
[23] Christ M. On almost-everywhere convergence of Bochner-Riesz means in higher dimensions.Proc. Amer. Math. Soc., 1985, 95(1): 16-20
[24] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonolity and OscillatoryIntegrals. Princeton Univ. Press, Princeton, New Jersey, 1993, 10-42
[25] Hu G, Lu S. The commutator of the Bochner-Riesz operator. T?ohoku Math. J., 1996, 48(2):259-266
[26] Hu G, Lu S. The maximal operator associated with the commutator of the Bochner-Rieszoperator. Beijing Math., 1996, 2(1): 96-106
[27] Hu G, Lu S. A weighted L2 estimates for the commutator of the Bochner-Riesz operator.Proc. Amer. Math. Soc., 1997, 125(10): 2867-2873
[28] Ma B, Hu G. Maximal operators associated with commutators of spherical means. TohokuMath. J., 1998, 50(3): 349-363
[29] Hu G, Lu S, Ma B. Commutator of convolution operator. Acta. Math. Sinica, 1999, 42(2):359-368
[30] Hu G, Ma B. L2(Rn) Boundedness for the Commutators of Homogeneous Singular IntegralOperators. Acta. Math. Sinica, 2004, 20(1): 785-792
[31] Stein E M. Singular Integrals and Di?erentiability Properties of Functions. Princeton Univ.Press, Princeton, NJ, 1970, 31-40
[32] Paluszynski M. Characterization of the Besov spaces via the commutator operator of Coif-man, Rochberg and Weiss. Indiana Univ. Math. J., 1995, 44(1): 1-17
[33] Liu Z. The Lipschitz estimats for the maximal commutator of the Bochner-Riesz operator.Beijing Normal Univ., 2002, 38(2): 160-164
[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 Triebel-Lizorkin spaces. Integr Equ. Oper.Theory, 2004, 49(1): 65-75
[36] Xia X. Boundedness of some singular integral operators and their commutators: Ph. D.Thesis Beijing Normal Univ. Beijing: Beijing Normal Univ., 2007, 10-28
[37] Tomas P. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc., 1975,81(2): 477-478
[38] Liu L X, Ma B L. The commutators of the multiplier operators. to appear Pan. Amer.Math., 2008, 23(2): 45-56
[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] Shi X, Sun Q. Weighted norm inequalities for Bochner-Riesz operators and singular integraloperators. Proc. Amer. Math. Soc., 1992, 116(3): 665-673
[42] Carbery A, Rubio de Ftancia J L, Vega L. Almost everywhere summability of Fourierintegrals. J. LondonMath. Soc., 1998, 38(33): 513-524
[43] Cordoba A. The Kakeya maximal function and the spherical summation multiplier. Amer.J. Math., 1977, 99(1): 1-22
[44] Devore R A. Sharpley R C. Maximal functions measuring smoothness. Mem. Amer. Math.Soc., 1984, 47(293): 163-174
[45] Janson S, Taibleson M, Weiss G. Elementary characterization of the Morrey–Campanatospaces. Lect. Notes in Math., 1983, 992(1): 101-114
[46] Stein E M. Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton univ.Press, Princeton, New Jersey, 1970, 15-25
[47] Ho?man S. Weighted norm inequalities and vector-valued inequalities for certain roughoperators. Indiana Univ. Math. J., 1993, 42(1): 1-14
[48] Bergh J, L¨ofstro¨m J. Interpolation Spaces-an Introduction. Springer-Verlag, New York,1976, 58(223): 23-49
[2] Calderon A P. Cauchy integrals on Lipschitz curves and related operators. Proc. Nat. Acad.Sci. USA., 1977, 74(4): 1324-1327
[3] John F, Nirenberg L. On functions of bounded mean oscillation. Comm. Pure Appl. Math.,1961, 14(3): 415-426
[4] Coifman R, Rochberg R, Weiss G. Facorization theorems for Harday spaces in several vari-ables. Ann. of Math., 1976, 103(3): 611-635
[5] Chanillo S. A note on commutators. Indiana Unv. Math. J., 1982, 31(1): 7-16
[6] Janson S. Mean oscillation and commutaotrs of singular integral operators. Ark. Math.,1978, 16(1): 263-270
[7] Fe?erman C. Stein E M. Hp spaces of several variables. Acta. Math., 1972, 129(1): 137-193
[8] Janson S. Mean oscillation and of singular integral operators. Ark. Math. J., 1982, 31(1):7-16
[9] Perez C. Endpoint estimates for commutators of singular integral operators. J. Funct. Anal.,1995, 128(1): 163-185
[10] Muckenhoupt B. Weighted norm inequalities for the Harday maximal function. Trans. Amer.Math. Soc., 1972, 165(1): 207-226
[11] Kurtz D S. Littlewood-Paley and multiplier theorems on weighted Lp spaces. Trans. Amer.Math. Soc., 1980, 209(1): 235-254
[12] Chanillo S. Weighted norm inequalities for strongly singular convolution operators. Trans.Amer. Math. Soc., 1984, 281(1): 77-107
[13] Bloom S. A commutator theorem and weighted BMO. Trans. Amer. Math. Soc., 1985,292(1): 103-122
[14] You Z. Results on commutators obtained from weighted norm inequalities. Chinese Adv. inMath., 1988, 17(1): 79-84
[15] Ga′rcia-Cuerva J, Harboure E, Segovia C, Torrea J L. Weighted norm inequalities for com-mutators of strongly singular integrals. Indiana Univ. Math. J., 1991, 40(44):1397-1420
[16] Alvarez J, Bagby R J, Kurtz D S, P′erez C. Weighted estimates for commutators of linearoperators. Studia Math., 1993, 104(1): 195-209
[17] Rubio de Francia J L. Maximal functions and Fourier transforms. Duke Math J., 1986,53(2): 395-405
[18] Duoandikoetxea J, Rubio de Francia J L. Maximal and singular integral operators viaFourier transform estimates. Invent Math., 1996, 84(3): 541-561
[19] Hirschman I I. Multiplier transformations. Duke Math. Jour., 1959, 26(2): 222-242
[20] Carleson L. Sj¨olin P, Oscillatory integrals and a multiplier problem for the disc. StudiaMath., 1972, 44(3): 287-299
[21] Fe?ernan C. A note on the spherical summation multiplier. Israel J. Math., 1973, 15(1):44-52
[22] Carbery A. The boundedness of the maximal Bochner-Riesz operator on L4(R2). DukeMath. J., 1983, 50(2): 409-416
[23] Christ M. On almost-everywhere convergence of Bochner-Riesz means in higher dimensions.Proc. Amer. Math. Soc., 1985, 95(1): 16-20
[24] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonolity and OscillatoryIntegrals. Princeton Univ. Press, Princeton, New Jersey, 1993, 10-42
[25] Hu G, Lu S. The commutator of the Bochner-Riesz operator. T?ohoku Math. J., 1996, 48(2):259-266
[26] Hu G, Lu S. The maximal operator associated with the commutator of the Bochner-Rieszoperator. Beijing Math., 1996, 2(1): 96-106
[27] Hu G, Lu S. A weighted L2 estimates for the commutator of the Bochner-Riesz operator.Proc. Amer. Math. Soc., 1997, 125(10): 2867-2873
[28] Ma B, Hu G. Maximal operators associated with commutators of spherical means. TohokuMath. J., 1998, 50(3): 349-363
[29] Hu G, Lu S, Ma B. Commutator of convolution operator. Acta. Math. Sinica, 1999, 42(2):359-368
[30] Hu G, Ma B. L2(Rn) Boundedness for the Commutators of Homogeneous Singular IntegralOperators. Acta. Math. Sinica, 2004, 20(1): 785-792
[31] Stein E M. Singular Integrals and Di?erentiability Properties of Functions. Princeton Univ.Press, Princeton, NJ, 1970, 31-40
[32] Paluszynski M. Characterization of the Besov spaces via the commutator operator of Coif-man, Rochberg and Weiss. Indiana Univ. Math. J., 1995, 44(1): 1-17
[33] Liu Z. The Lipschitz estimats for the maximal commutator of the Bochner-Riesz operator.Beijing Normal Univ., 2002, 38(2): 160-164
[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 Triebel-Lizorkin spaces. Integr Equ. Oper.Theory, 2004, 49(1): 65-75
[36] Xia X. Boundedness of some singular integral operators and their commutators: Ph. D.Thesis Beijing Normal Univ. Beijing: Beijing Normal Univ., 2007, 10-28
[37] Tomas P. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc., 1975,81(2): 477-478
[38] Liu L X, Ma B L. The commutators of the multiplier operators. to appear Pan. Amer.Math., 2008, 23(2): 45-56
[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] Shi X, Sun Q. Weighted norm inequalities for Bochner-Riesz operators and singular integraloperators. Proc. Amer. Math. Soc., 1992, 116(3): 665-673
[42] Carbery A, Rubio de Ftancia J L, Vega L. Almost everywhere summability of Fourierintegrals. J. LondonMath. Soc., 1998, 38(33): 513-524
[43] Cordoba A. The Kakeya maximal function and the spherical summation multiplier. Amer.J. Math., 1977, 99(1): 1-22
[44] Devore R A. Sharpley R C. Maximal functions measuring smoothness. Mem. Amer. Math.Soc., 1984, 47(293): 163-174
[45] Janson S, Taibleson M, Weiss G. Elementary characterization of the Morrey–Campanatospaces. Lect. Notes in Math., 1983, 992(1): 101-114
[46] Stein E M. Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton univ.Press, Princeton, New Jersey, 1970, 15-25
[47] Ho?man S. Weighted norm inequalities and vector-valued inequalities for certain roughoperators. Indiana Univ. Math. J., 1993, 42(1): 1-14
[48] Bergh J, L¨ofstro¨m J. Interpolation Spaces-an Introduction. Springer-Verlag, New York,1976, 58(223): 23-49