Harmonic Analysis for Orthogonal Expansions

我们的研究成果总结如下。首席研究员Kanjin得到了关于Jacobi展开式的Hardy不等式。经典的Hardy不等式是单位圆盘上Hardy空间中函数的傅里叶系数的最常见的不等式之一。这个不等式最初是用复数方法证明的。但是，将复数方法应用于正交展开式的研究是困难的。我们的想法是利用实Hardy空间的理论，特别是实Hardy空间的原子分解刻画，它允许我们讨论关于正交展开式的不等式问题。将原子分解应用于雅可比函数系，证明了我们的Hardy不等式。此外，他还研究了Hankel变换的移植算子，得到了算子在实Hardy空间上的有界性。土屋研究了具有Ventcel-Visik边界条件的扩散方程，并构造了其经典解。Ichinose研究了非对易谐振子的谱Zeta函数，得到了Zeta函数作为亚纯函数对整个复平面具有解析延拓的结果。Sato考虑了具有非齐次核的奇异积分，并在一定的附加条件下得到了这类奇异积分的加权弱型(1，1)估计。Tohge得到了一些全纯曲线极值于Cartan亏损关系的例子。

Our research results are summarized as follows. The head investigator Kanjin has obtained Hardy's inequality with respect to the Jacobi expansions. The classical Hardy's inequality is one of the most familiar inequalities on the Fourier coefficients of functions in the Hardy space on the unit disc. The inequality was originally proved by complex method. But, it is difficult to apply complex method to the study of orthogonal expansions. Our idea is to use the real Hardy space theory, especially the atomic decomposition characterization of the real Hardy space, and it allows us to discuss problems on inequalities with respect to orthogonal expansions. Our Hardy's inequality has proved by applying the atomic decomposition to the Jacobi function system. Further, he has studied the transplantation operators for the Hankel transform, and he obtained the boundedness of the operators on the real Hardy space.The investigator Tsuchiya has investigated the diffusion equations with Ventcel'-Visik boundary conditions, and constructed their classical solutions. Ichinose has studied the spectral zeta function for the non-commutative harmonic oscillator and gotten the result that the zeta function has an analytic continuation to the whole complex plane as a meromorphic function. Sato has considered singular integrals with non-homogeneous kernels, and under certain additional conditions he obtained the weighted weak type (1,1) estimates for such singular integrals. Tohge has gotten some examples of holomorphic curves extremal to Cartan's defect relation.