算子交换子的有界性和紧性

Because the commutator generated by integral operator and BMO function has an important application in partially differential equation. Many important operators gave a characterization of BMO space. We give the characterization of BMO space by studying the boundedness of the commutator of the Marcinkiewicz integral operators on Morrey space. .It is well known that Littlewood-Paley operators, such as the Lusin area integral and Littlewood-Paley function play very important roles in harmonic analysis and PDE. Therefore, it is a very interesting problem to discuss the properties of the commutators for the Littlewood-Paley operators. We study the characterization of the boundedness and compactness of the commutators of Littlewood-Paley operators on Morrey space. .The Riesz potential is in connection with Sobolev space and is found important applications to the the restriction theorem for the Fourier transform and Fourier integral operator. We consider the characterization of the compactness of the commutators of the commutators of fractional integrals on Morrey space. .We focuse on four contents: the characterization of the boundedness and compactness of the commutators of Littlewood-Paley operators on Morrey space; the results about the characterization of the compactness of the commutators of fractional integrals on Morrey space; the theories of the commutators of parabolic Marcinkiewicz and parabolic Littlewood-Paley operators on Morrey space.

由于交换子与偏微分方程、拟微分算子、Cauchy 型积分的密切关系，同时其本身也是很典型的非卷积型的Calderon-Zygmund 算子，因此一直引起人们的兴趣。带BMO 函数的奇异积分交换子在偏微分方程上的重要应用, 所以奇异积分交换子是继奇异积分之后引起人们极大研究兴趣的一种重要算子。因此, 自20世纪70 年代以来, 对 Coifman-Rochberg-Weiss 型交换子的研究十分活跃并取得了非常丰硕的成果。我们主要研究Littlewood-Paley算子的交换子的有界性有紧性特征；分数次积分算子交换子的紧性特征；抛物型Marcinkiewicz积分算子交换子以及抛物型Littlewood-Paley积分交换子的有界性特征；高阶交换子的有界性及非线性算子的有界性。

由BMO函数生成的奇异积分交换子在偏微分方程中有重要的作用，故在学者们研究了奇异积分算子之后，奇异积分交换子引起了学者们的兴趣。项目主要研究的是算子及交换子的有界性和紧性。特别是针对高阶奇异积分算子生成的交换子，我们研究了它在Morrey空间上的有界性。由于Marcinkiewicz积分算子是次线性算子，需要将积分区域分成两部分进行估计。而Lilltewood-Paley积分算子生成的交换子估计更为复杂，需要将积分区域分成更多部分进行估计。. 我们的主要研究内容有以下三个部分： Littlewood-Paley算子交换子在Morrey空间上的有界性和紧性；分数次积分算子生成的交换子在Morrey空间上的有界性和紧性；抛物型Littlewood-Paley 算子交换子在Morrey空间上的有界性和紧性。