| Research Abstract |
In this research we investigated the usage of fractional integrals in real analysis and also in harmonic analysis on semisimple Lie groups. E. Nakai, the investigator, studied boundedness of the fractional integral operators on various function spaces, especially, he generalized the operators and function spaces. T. Kawazoe, the head investigator, applied the theory of fractional integrals to construct real Hardy spaces on semisimple Lie groups. In this sense, we attained our aim of this research -in harmony with representation theory and real analysis.
The theory of real Hardy spaces was generalized on the so-called space of homogeneous type satisfying the doubling condition, however, little works were done on the space of non-homogeneous type. In this research we took real rank one semisimple Lie groups G as an example of the space of non-homogeneous type, and introduced real Hardy spaces for K-bi-invariant functions. This corresponds to harmonic analysis on Fourier-Jaccobi transforms
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and on the weight theory with exponential growth order. We first, by using the Abel transform, reduced the Fourier-Jaccobi analysis to Euclidean one on the real line R. Since the Abel transform was written by fractional integrals, the method of real analysis related to fractional integrals were useful in this analysis. Next, five extended the theory of fractional integrals and derivatives and then characterized real Hardy spaces defined by maximal functions and atoms. Our main theorem is the following. Let W be the modified Abel transform with Exp(ρx)-multiplication and V the inverse of W. Then, we see H1 (G//K) ⊂V (H1(R)), where H1(G//K) and HI(R) are respectively the real Hardy spaces on G and R defined by the radial maximal functions. Moreover, let A1(G//K)+ denote the atomic Hardy space on G defined by restricting the moment condition of atoms only for ones with redius <1. Then it follows that H1(G//K)= V(H1(R))∩A1(G//K)+. In the proof we applied some methods in real analysis related to fractional integrals and atomic decompositions. Less
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