Anew development of real Hardy spaces in non-commutative harmonic analysis-a fusion of representation theory, real analysis, and probability

在这四年的研究中，我们在非对易调和分析中对真实的哈代空间的刻画及其应用方面取得了重要的成果。在第一阶段，我们的目标是真实的秩1半单李群G上的K-双不变函数，然而，它扩展到詹比分析，并进一步扩展到Sturm-Liouville超群，我们成功地得到了用径向极大函数定义的真实的Hardy空间H^1（Δ）与经典的真实的哈代空间H^1 P（R）之间的关系。这个关系是从阿贝尔变换的积分表达式得出的，阿贝尔变换是由分数次积分给出的。其关键思想是用普通分数阶导数来表示阿贝尔变换的逆。作为H^1（Δ）的应用，我们考虑了Poisson极大算子、Littlewood-Paley g-函数和Lusin面积函数S的（H^1（Δ），L^1（Δ））有界性.众所周知，当p>1时，这些算子在L^P（Δ）上有界，但当p=1时，我们没有结果。因此我们的（H^1（Δ），L1（Δ））有界性是新的和有意义的。在证明中，我们使用了上述H1（Δ）的特征，并将非交换调和分析中的论点简化为欧几里德分析。在这个过程中，我们期望积分算子在欧几里德情形下具有相同的性质。然而，在我们的研究中，我们注意到由非切向积分定义的Lusin面积算子S的（H1^（Δ），L^<1（Δ）>）有界性依赖于非切向区域的形状，特别是区域的角度.这个结果是基于Δ具有指数增长阶的事实。这种现象是独特的，因此非常有趣。

In these 4 years research we have obtained significant results on a characterization of real Hardy spaces in non-commutative harmonic analysis and their applications. In the first stage our target was K-bi-invariant functions on real rank one semisimple Lie groups G, however, it extended to Jambi analysis, and further to Sturm-Liouville hypergroups.We succeed to obtain a relation between the real Hardy space H^1(Δ), which is defined by using a radial maximal function, and the classical real Hardy space H^1P(R). This relation follows from an integral expression of the Abel transform, which is given by fractional integrals. The key idea is to express the inverse of the Abel transform in terms of ordinary fractional derivatives. This result is also useful to analyze boundedness of some integral operators.As an application of H^1(Δ), we consider (H^1(Δ), L^1(Δ)) boundedness of the Poisson maximal operator, the Littlewood-Paley g-function and the Lusin area function S. It is well-known that these operators are bounded on L^P(Δ) for p>1, however we have no results in the case of p=1. Hence our (H^1(Δ), L1(Δ)) boundedness is new and significant. In the proof we use the characterization of H1(Δ) stated above and reduce the arguments in non-commutative harmonic analysis to Euclidean analysis. In this process we expect that integral operators have the same properties in the Euclidean case. However, in our research we notice that the (H1^(Δ), L^<1(Δ)>) boundedness of the Lusin area operator S, which is defined by using a non-tangential integral, depends on the shape of the non-tangential domain, especially the angle of the domain. This result is based on the fact that Δ has an exponential growth order. This phenomenon is unique and therefore, is quite interesting.