A study of harmonic analysis for orthogonal expansions

本课题的主要研究成果总结如下。在实Hardy空间上证明了Hankel变换的移植定理。移植算子是将正交系中具有傅里叶展开的函数映射到相对于另一正交系具有相同傅立叶系数的函数的算子。移植定理是断言移植算子的有界性的定理。这类定理在正交展开的调和分析中是一个有用的工具。汉克尔变换是积分变换的一种，作为特例与傅里叶变换重合。实Hardy空间上算子的估计使得我们可以得到Lebeegue空间上算子的相应估计。在这样一个有用的方案中，我们得到了一个移植定理。移植算子被认为是希尔伯特变换的推广。众所周知，希尔伯特变换将一个具有一定条件的函数映射到一个可积函数。我们证明了Hankel变换的移植算子具有相同的性质。利用这一结果，我们证明了Hankel变换的Cesaro算子在可积函数空间和实Hardy空间上是有界的，得到了积分变换型的Paley不等式。经典的Paley不等式说，在实Hardy空间中函数的傅里叶展开中，其在Hadamard间隙上的傅立叶系数的绝对值之和收敛，并且该和有界于该函数的实Hardy空间范数的平方。我们已经证明了与经典的Paley不等式相同类型的不等式对于Hankel变换也成立。

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.