2006 Fiscal Year Final Research Report Summary
A study of harmonic analysis for orthogonal expansions
Project/Area Number 
17540155

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Kanazawa University 
Principal Investigator 
KANJIN Yuichi Graduate School of Natural Science and Technology., Professor, 自然科学研究科, 教授 (50091674)

CoInvestigator(Kenkyūbuntansha) 
SATO Shuichi Faculty of Education, Associate Professor, 教育学部, 助教授 (20162430)
TOHGE Kazuya Graduate School of Natural Science and Technology, Associate Professor, 自然科学研究科, 助教授 (30260558)
ARAI Hitosi The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理科学研究科, 教授 (10175953)
MIYACHI Akihiko Tokyo Woman's Christian University, College of Arts and Science, Professor, 文理学部, 教授 (60107696)

Project Period (FY) 
2005 – 2006

Keywords  transplantation operator / Cesaro operator / Paley's inequality / real Hardy space / Hankel transform 
Research Abstract 
Our main results of this research project are summarized as follows. The transplantation theorem for the Hankel transform has been proved on the real Hardy space. A transplantation operator is an operator which maps a function with the Fourier expansion in an orthogonal system to the function with the same Fourier coefficients with respect to another orthogonal system. A transplantation theorem is a theorem which asserts the boundedness of the transplantation operator. This type of theorem is a useful tool in harmonic analysis for orthogonal expansions. The Hankel transform is one of the integral transforms, and coincides with the Fourier transform as a special case. Estimations of operators on the real Hardy space allow us to get the corresponding estimations of the operators on the Lebesegue spaces. In such a useful scheme, we have obtained a transplantation theorem. Transplantation operators are regarded as a generalization of the Hilbert transform. It is known that the Hilbert transform maps a function with certain conditions to an integrable function. We have proved that the transplantation operators for the Hankel transform have the same properties. Using this result, we have showed that the Cesaro operators for the Hankel transform are bounded on the space of integrable functions and on the real Hardy space. We have obtained Paley's inequality of integral transform type. The classical Paley inequality says that in the Fourier expansion of a function in the real Hardy space, the sum of the absolute values of its Fourier coefficients taken over the Hadamard gaps converges, and the sum is bounded by the square of the real Hardy space norm of the function. We have showed that an inequality of the same type as the classical Paley inequality holds for the Hankel transform.

Research Products
(10 results)