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Harmonic Analysis for Some Orthogonal Expansions

一些正交展开的调和分析

基本信息

批准号：
13640160
负责人：
KANJIN Yuichi
金额：
$ 1.86万
依托单位：
Kanazawa University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640160/
关键词：
Hardy space Paley's inequality Hausdorff operator diffusion equation the self-adjoint Trotter-Kato product formula Marcinkiewicz integral Riccati differential equation Hardy's ineguality Paley's inequality Hausdorff operatol Raly's ine

项目摘要

Our research results are summarized as follows. The head investigator Kanjin has obtained Paley's inequality and Hardy's inequality with respect to the Jacobi expansions. The classical Peley's inequality and Hardy's inequality are two of the most familiar inequalities on the Fourier coefficients of functions in the Hardy space of certain analytic functions in the unit disc. The inequalities were originally proved by complex method. It is difficult to study orthogonal expansions by using complex method. Recent development of the real Hardy space theory, especially the atomic decomposition characterization or the real Hardy space and BMO space duality, allows to discuss problems on inequalities with respect to orthogonal expansions. Our Hardy's inequality have proved by applying the atomic decomposition to the Jacobi function system and Our Paley's inequality has gotten by using the real Hardy space and BMO space duality. Further, he has studied the Cesaro operator and, generally, the Hausdorff operator. The result says that the Hausdorff operator is bounded on the real Hardy space with parameter p smaller than one under some conditions.The investigator Tsuchiya has investigated convergence of Dirichlet forms of diffusion process without assuming that the underlying measures are fixed or compatible with a fixed one. Ichinose has obtained more results on the self-adjoint Trotter-Kate product formula with operator norm. Sato has considered Marcinkiewicz integrals arising from rough kernels satisfying LlogL condition on the unit (n-l)-sphere and proved the weak type (1,1) estimates. Tohge has studied a Riccati differential equation whose coefficient is expressible in terms of a special Weierstrass pe-function and shown that all the solutions are meromorphic.

我们的研究结果总结如下。研究者Kanjin得到了关于Jacobi展开式的Paley不等式和哈代不等式。经典的Peley不等式和哈代不等式是单位圆上解析函数在哈代空间中Fourier系数的两个最常见的不等式。这些不等式最初是用复形法证明的。用复形法研究正交展开式比较困难。真实的Hardy空间理论的最新发展，特别是真实的哈代空间和BMO空间对偶的原子分解特征，使得讨论关于正交展开式的不等式问题成为可能。利用Jacobi函数系的原子分解证明了我们的哈代不等式，利用真实的哈代空间和BMO空间对偶得到了我们的Paley不等式.此外，他研究了塞萨罗运营商，一般来说，豪斯多夫运营商。结果表明，在一定条件下，Hausdorff算子在参数p小于1的真实的哈代空间上是有界的。Tsuchiya研究了扩散过程Dirichlet形式的收敛性，而不假设基础测度是固定的或与固定测度相容。Ichinose在算子范数下的自伴Trotter-Kate乘积公式上得到了更多的结果。Sato在单位（n-l）球面上考虑了由满足LlogL条件的粗糙核所产生的Marcinkiewicz积分，并证明了其弱（1，1）型估计. Tohge研究了一个Riccati微分方程，其系数可用一个特殊的Weierstrass函数表示，并证明了该方程的所有解都是亚纯的。