两类交换子的端点估计
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摘要
本文共分四章,主要介绍和讨论了如下几个内容：拟微分算子的交换子在Hardy空间的有界性和紧性；非双倍测度下,带参数的Marcinkiewicz奇异积分算子的交换子在Lebesgue空间,Hardy空间和RBMO空间有界性.全文内容安排如下：
第一章首先介绍了奇异积分算子及其交换子的发展历史和现状,特别对拟微分算子和Marcinkiewicz奇异积分算子及其交换子的研究背景和现状做了详细的介绍.然后针对这两个算子的交换子研究现状提出了五个问题.最后简要的介绍了我们得到的主要结果.
第二章讨论了象征(?)(χ,ξ)属于象征类S01,δ(0≤δ<1),当b分别在BMO函数空间或BMO∞时,对应于此象征的交换子T(?)b是H1(Rd)到L1(Rd)的有界算子的充分必要条件是b∈LMO或b∈LMO∞.
第三章给出了象征(?)(χ,ξ)属于象征类S01,δ5(0≤δ占<1)时,交换子T(?)b是H1(Rd)到L1(Rd)的紧算子的充分条件为b∈CLMO或b∈CLMO∞.
第四章证明了在非齐次型空间上,若b属于Lipschitz空间Lipβ(μ)(0<β≤1), Marcinkiewicz积分核满足某种Hormander条件,由b和带参数的Marcinkiewicz积分算子(?)(?)生成的交换子3(?)b在Lp(μ)(1In this dissertation, we focus on the boundedness or compactness for commutator of cer-tain pseudo-differential operators and parametrized Marcinkiewicz singular integral operators. It comprises four chapters.
In chapter1, we give a survey on the background and recent development of general sin-gular integral operators and their commutator. In particular, we gather some notations about pseudo-differential operators, Marcinkiewicz singular integral operators and their commuta-tors. Finally, we simply list our work.
In chapter2, we discuss endpoint estimates for the commutator of certain pseudo-differential operators. Let σ(x,ξ)∈S01δ(0≤δ<1),b∈BMO(b∈BMO∞), then the commutator [b, Tσ] is bounded from H1(Rn) into L1(Rn) if and only if b∈LMO(b∈LMO∞).
In chapter3, we show two sufficient condition which the commutator [b,Tσ] is com-pacted operators. If b∈CLMO or b∈CLMO∞, then the commutator [b,Tσ] is compacted from H1(Rn) into L1(Rn) for any σ{x,ξ)∈01δ^(0≤δ<1).
In chapter4, we deal with the endpoint estimates for the commutator of parametrized Marcinkiewicz singular integral operators. On non-double measure space, if b∈Lipβ(μ)(0<β≤1) and Marcinkiewicz integral kernel satisfy some condition, then the parametrized Marcinkiewicz commutators M(?)b is bounded on Lebesgue spaces L/?(μ)(1
第一章首先介绍了奇异积分算子及其交换子的发展历史和现状,特别对拟微分算子和Marcinkiewicz奇异积分算子及其交换子的研究背景和现状做了详细的介绍.然后针对这两个算子的交换子研究现状提出了五个问题.最后简要的介绍了我们得到的主要结果.
第二章讨论了象征(?)(χ,ξ)属于象征类S01,δ(0≤δ<1),当b分别在BMO函数空间或BMO∞时,对应于此象征的交换子T(?)b是H1(Rd)到L1(Rd)的有界算子的充分必要条件是b∈LMO或b∈LMO∞.
第三章给出了象征(?)(χ,ξ)属于象征类S01,δ5(0≤δ占<1)时,交换子T(?)b是H1(Rd)到L1(Rd)的紧算子的充分条件为b∈CLMO或b∈CLMO∞.
第四章证明了在非齐次型空间上,若b属于Lipschitz空间Lipβ(μ)(0<β≤1), Marcinkiewicz积分核满足某种Hormander条件,由b和带参数的Marcinkiewicz积分算子(?)(?)生成的交换子3(?)b在Lp(μ)(1
In chapter1, we give a survey on the background and recent development of general sin-gular integral operators and their commutator. In particular, we gather some notations about pseudo-differential operators, Marcinkiewicz singular integral operators and their commuta-tors. Finally, we simply list our work.
In chapter2, we discuss endpoint estimates for the commutator of certain pseudo-differential operators. Let σ(x,ξ)∈S01δ(0≤δ<1),b∈BMO(b∈BMO∞), then the commutator [b, Tσ] is bounded from H1(Rn) into L1(Rn) if and only if b∈LMO(b∈LMO∞).
In chapter3, we show two sufficient condition which the commutator [b,Tσ] is com-pacted operators. If b∈CLMO or b∈CLMO∞, then the commutator [b,Tσ] is compacted from H1(Rn) into L1(Rn) for any σ{x,ξ)∈01δ^(0≤δ<1).
In chapter4, we deal with the endpoint estimates for the commutator of parametrized Marcinkiewicz singular integral operators. On non-double measure space, if b∈Lipβ(μ)(0<β≤1) and Marcinkiewicz integral kernel satisfy some condition, then the parametrized Marcinkiewicz commutators M(?)b is bounded on Lebesgue spaces L/?(μ)(1
引文
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[23]Y. Ding, S. Lu, and Q. Xue. On Marcinkiewicz integral with homogeneous kernels. Journal of Mathematical Analysis and Applications, 245(2):471-488,2000.
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[25]Y. Ding, S. Lu, and K. Yabuta. A problem on rough parametric Marcinkiewicz functions. Journal of'the Australian Mathematical Society, 72(1):13-22, 2002.
[26]Y. Ding, S. Lu, and P. Zhang. Weighted weak type estimates for commutators of the Marcinkiewicz integrals. Science in China Series A: Mathematics, 47(1):83-95,2004.
[27]D. Fan and S. Sato. Weak type (1,1) estimates for Marcinkiewicz integrals with rough kernels. Tohoku Mathematical Journal,53(2):265-284,2001.
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[36]G. Hu, H. Lin, and D. Yang. Marcinkiewicz integrals with non-doubling measures. Integral Equations and Operator Theory, 58(2):205-238,2007.
[37]G. Hu, Y. Meng, and D. Yang. Multilinear commutators of singular integrals with non doubling measures. Integral Equations and Operator Theory, 51(2):235-255,2005.
[38]G. Hu and D. Yan. On the commutator of the Marcinkiewicz integral. Journal of mathematical analysis and applications, 283(2):351-361,2003.
[39]M. Huixia and L. Shanzhen. Boundedness of generalized higher commutators of Marcinkiewicz integrals. Acta Mathematica Scientia, 27(4):852-866,2007.
[40]S. Janson. Mean oscillation and commutators of singular integral operators. Arkiv for Matematik, 16(1):263-270,1978.
[41]F. John and L. Nirenberg. On functions of bounded mean oscillation. Communications on Pure and Applied Mathematics, 14(3):415-426,1961.
[42]J. Journe. Calderon-Zygmund operators, pseudo-differential operators, and the Cauchy integral of Calderon. Lecture Notes in Mathematics, 1983.
[43]Y. Karlovich. Algebras of pseudo-differential operators with discontinuous symbols. Modern Trends in Pseudo-Differential Operators, 172:207-233,2007.
[44]Y. I. Karlovich. An algebra of pseudo-differential operators with slowly oscillating symbols. Proceedings of the London Mathematical Society,92(3):713-761,2006.
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