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

在这4年的研究中，我们对非交换调和分析中实Hardy空间的刻画及其应用取得了重要的成果。在第一阶段，我们的目标是实秩1半单李群G上的k -双不变函数，然而，它扩展到Jambi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实Hardy空间H^1（Δ）与经典实Hardy空间H^1P(R)之间的关系。这个关系来源于阿贝尔变换的积分表达式，它是由分数积分给出的。关键思想是用普通分数阶导数来表示阿贝尔变换的逆。该结果对分析某些积分算子的有界性也有帮助。作为H^1（Δ）的应用，我们考虑了泊松极大算子、Littlewood-Paley g函数和Lusin区域函数s （H^1（Δ）、L^1（Δ））的有界性。众所周知，对于P >1，这些算子在L^P（Δ）上是有界的，但是在P =1的情况下我们没有结果。因此我们的（H^1（Δ）， L1（Δ））有界性是新的和重要的。在证明中，我们使用上述H1（Δ）的表征，并将非交换调和分析中的论点简化为欧几里得分析。在这个过程中，我们期望积分算子在欧几里得情况下具有相同的性质。然而，在我们的研究中，我们注意到Lusin区域算子S的（H1^（Δ）， L^<1（Δ）>）有界性取决于非切域的形状，特别是域的角度，这是用非切积分定义的。这个结果是基于Δ具有指数增长顺序的事实。这种现象很独特，因此也很有趣。

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.