THE RESEARCH OF OPERATORS ON LORENTZ SPACES BY THE METHOD OF HARMONIC ANALYSIS

本研究的目的是用谐子分析的方法研究洛伦兹空间上算子的性质。我们的调查在每个特殊的点都做了研究。内容如下：Sato研究了（0,pi）上的一些Hankel变换算子与Jacobi正交系统算子之间的关系。他还研究了一个与Lorentz-Zygmund空间上的算子有关的不等式。Mori研究了代数非退化亚纯映射的构造问题和与亚纯函数相关的唯一性问题。Mizuhara利用广义Morrey函数、块和Riesz势证明了H^1函数的弱分解定理。应用这一结果，他还观察到广义Morrey空间上Reisz势和局部可积函数之间的对易子有界的必要性。Nakada研究了黎曼球上有理函数Julia集的一些几何性质。Kawamura讨论了关于混沌映射在测量空间上的概率密度函数行为的理论的推广。他研究了连接两个混沌映射的新型共轭性。Sekigawa利用M个明显变换的Clifford矩阵表示研究了高维欧几里德空间上的M个明显变换。Kanjin证明了Hardy空间上有关Hankel变换的移植算子的有界性。他还证明了Cesaro算子的有界性，以及Hardy空间中关于Jacobi级数的傅里叶系数的Hardy不等式。

Our purpose of this research is to study the properties of operators on Lorentz spaces by the method of haomonic analysis. Our investigations did the research at each special point. The content is as follows :Sato studied the relation between some operators of Hankel transforms and the operators of Jacobi orthgonal system on (0,pi). Also he studied an inequality related to the operators on the Lorentz-Zygmund spaces. Mori researched about a problem of constructions of algebraically nondegenereate meromorophic mappings and the uniquness problem related to meromorphic functions. Mizuhara showed the weak factorization theorem of H^1-functions due to generalized Morrey functions, blocks and the Riesz potential. Also applying this result, he observed the necessity for which the commutator between the Reisz potential and a locally integrable function to be bounded on the generalized Morrey spaces. Nakada studied some geometric properties of the Julia set of rational functions on the Riemann sphere. Kawamura discussed an generalization of the theory concerning the behavior of probability density functions associated with chaotic maps on a measure space. He studied conjugacy of new type connecting two chaotic maps. Sekigawa studied the M"obious transformations on the high dimensional Euclidean spaces, by using Clifford matrix representations of M"obious transformations. Kanjin showed the boundedness of the transplantation operators concerning the Hankel transform on the Hardy spaces. Also he showed the boundedness of Cesaro operators, and Hardy's inequality related to the Fourier coefficients with respect to the Jacobi series on the Hardy's space.