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

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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.