Harmonic Analysis on Orthogonal Expansions

我们的研究成果总结如下。首席研究员Kanjin得到了关于Hermite和Laguerre展开式的Hardy不等式。经典的Hardy不等式是关于单位圆盘上某些解析函数的Hardy空间中函数的傅里叶系数的一个著名的不等式。这个不等式最初是用复数方法证明的。用复数方法研究正交多项式的展开式比较困难。实Hardy空间理论的最新发展，特别是实Hardy空间的原子分解刻画，使得讨论关于正交展开式的不等式问题成为可能。通过将原子分解应用于Hermite和Laguerre系统，证明了我们的不等式。利用我们的方法，我们还得到了Hankel变换的Hardy不等式。此外，我们还研究了离散Hardy空间，得到了该空间的分子刻画。作为应用，我们证明了离散Hardy空间的分数次积分定理和Marcinkiewicz型乘子定理，Ichinose证明了具有算子范数的Lie-Trotter-Kato乘积公式。Tohge得到了关于经典的Nevanlinna理论与线性微分方程组之间关系的一个结果。Tsuhiya在系数的Holder条件下，证明了由两个区域上的两个扩散过程叠加得到的马尔可夫过程的Feller性质。Sato考虑了Al-权，证明了核满足Dini条件的振荡奇异积分的加权弱型(1，1)估计。

Our research results are summarized as follows. Head investigator Kanjin has obtained Hardy's inequalities with respect to the Hermite and Laguerre expansions. The classical Hardy's inequality is a well-known inequality on the Fourier coefficients of functions in the Hardy space of certain analytic functions in the unit disc. The inequality was originally proved by complex method. It is difficult to study the orthogonal polynomial expansions by using complex method. Recent development of the real Hardy space theory, especially the atomic decomposition characterization of the real Hardy space allows to discuss the problem on the inequalities with respect to the orthogonal expansions. Our inequalities have proved by applying the atomic decomposition to the Hermite and Laguerre systems. By our method we have also gotten Hardy's inequality for the Hankel transforms. Further, we have studied the discrete Hardy space and obtained the molecular characterization of the space. As its applications, we have proved the theorem of fractional integration and the Marcinkiewicz type multiplier theorem for the discrete Hardy space.Investigator Ichinose has proved the Lie-Trotter-Kato product formula with operator norm. Tohge has gotten a result on the relation between the classical Nevanlinna theory and linear differential equations. Tsuchiya has shown Feller property for a Markov process obtained by superposing two diffusion processes in two domains under Holder condition for coefficients. Sato has considered Al-weights and proved weighted weak type (1,1) estimates for oscillatory singular integrals with kernels satisfying a Dini condition.