HARMONIC ANALYSIS IN THE OPERATORS OF THE FUNCTION SPACES RELATED TO THE ORTHOGONAL SYSTEMS

本研究的目的是研究与正交系相关的函数空间的算子中的调和分析。我们的研究人员在每个特殊的时间点都进行了研究。Sato研究了关于Jacobi正交系到Hankel变换系以及Lorentz-Zygmund空间上的算子的一个转移定理的推广。Mori将Nevanlinna的四值定理推广到具有重数条件的偶集的情形。他还给出了在角域而不是整个复平面上满足条件的唯一值域的定理。最后，给出了亚纯函数及其导数的一个奇异值问题的奇异方向的存在定理。Mizuhara研究了广义Morrey空间上的奇异积分算子及其交换子，以及与加权Hardy算子有关的权函数。此外，他还研究了基努宁最近关于点…的结果Sobolev函数的Hardy-Littlewood极大函数的更精确估计。川村从概率论的角度研究了一维混沌映射的性质。特别是，他分析了由Perron-Frobenius算子在Hilbert空间上移动的概率密度函数的轨道，该轨道与混沌映射具有典型的联系。考虑了Hankel变换的移植算子。Kanjin证明了移植算子在一定条件下将一个可积函数映射到一个可积函数。作为应用，他得到了Hankel变换的Cesaro算子的L1有界性和H1有界性。Kanjin在可积函数空间的Hermite展开式和Laguerre展开式中建立了Hardy型不等式。Miyachi在欧式区域上引入了加权Hardy空间，并建立了这些空间的几个基本性质。利用他的加权Hardy空间，他将经典的Jacobi级数的移植定理推广到指数p小于或等于1的情形。他还证明了由超球级数定义的修正的Hilbert变换在他的加权Hardy空间中是有界的，并应用这一结果，把关于Hardy类的实变量刻画的Burkhold-Gundy-Silverstein定理推广到开区间上的加权Hardy类的情形。较少

Our purpose of this research is to study Harmonic Analysis in the operators of the function spaces related to the orthogonal systems. Our investigators did the research at each special point. The content is as followsSato studied the generalization of a transference theorem related to the operators from Jacobi orthogonal systems to Hankel transform systems, and the operators on Lorentz-Zygmund spaces. Mori generalized the Nevanlinna's four values theorem to the case of pair sets with conditions of multiplicities. He also gave theorems for unique range sets with conditions in an angular domain instead of the whole complex plane. Finally, he gave existence theorems of a singular direction for a shrare value problem of a meromorphic function and its derivative. Mizuhara studied some singular integal operators and its commutators on generalized Morrey spaces and some weight functions related to the weighted Hardy operators. Also, he studied recent Kinnunen's results concerning to the poinw … More ise estimates of the Hardy-Littlewood maximal function of a Sobolev function. Kawamura studied the property of one-dimensional chaotic maps in the view of probability theory. Especially he analyzed the orbit of a probability density function moved by the Perron-Frobenius operator on Hilbert space, which is canonically associated with chaotic map. The transplantation operators for the Hankel transform were considered. Kanjin have proved that the transplantation operator maps an integrable function under certain conditions to an integrable function. As an application, he obtained the L1-boundedness and H1-boundedness of Cesaro operators for the Hankel transform. Kanjin established Hardy-type inequalities in the Hermite and Laguerre expansions for the space of integrable functions. Miyachi introduced weighted Hardy spaces on a Euclidean domain and established several fundamental properties of these spaces. Using his weighted Hardy spaces, he extended classical transplantation theorem for Jacobi series to the case of the exponent p less than or equal to 1. He also proved that the modified Hilbert transform defined through ultraspherical series is bounded in his weighted Hardy spaces, and applying this result, he generalized the Burkholder-Gundy-Silverstein theorem concerning the real variable characterization of Hardy classes to the case of his weighted Hardy classes on the open interval. Less