A study of harmonic analysis for orthogonal expansions

本研究项目的主要成果总结如下。在真实的哈代空间上证明了Hankel变换的移植定理。移植算子是将具有正交系中的傅立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。一个移植定理是一个定理，断言移植算子的有界性。这类定理在正交展开式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，与傅里叶变换是一种特殊情况。利用真实的哈代空间上算子的估计，可以得到Lebesegue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，希尔伯特变换将具有某些条件的函数映射为可积函数。我们证明了Hankel变换的移植算子具有相同的性质。利用这个结果，我们证明了Hankel变换的塞萨罗算子在可积函数空间和真实的哈代空间上是有界的，并得到了积分变换型的Paley不等式。经典的Paley不等式说，在真实的Hardy空间中的函数的傅里叶展开中，其傅里叶系数在Hadamard间隙上的绝对值的和收敛，并且该和以函数的真实的哈代空间范数的平方为界。我们证明了一个与经典的Paley不等式相同类型的不等式对Hankel变换成立。

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.