Theory and applications of BMO and related function spaces on spaces of homogeneous type

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1. The theory of pointwise multipliers on the function space Lp is well known. In this research, we have developed this theory on several function spaces ; Lorentz, Orlicz, Morrey, BMO, and Campanato spaces. The pointwise multiplier is simple and basic. So we can examine properties of spaces of homogeneous type by studying it.2. The boundedness of singular integral operators and the Riesz potential are useful for the theory of partial differential equations and are studied by many authors. In particular, the boundedness of the Riesz potential (fractional integral) from L^p to L^q is well known as the Hardy-Littlewood-Sobolev theorem. In this research, we introduce generalized fractional integrals and extend this boundedness on several function spaces ; Orlicz, BMO_φ, Campanato, Morrey, weak-Orlicz and generalized Hardy spaces.3. We have developed the method to redefine the quasi-distance of the space of homogeneous type on which the function space is not change, and we have the following as applications :(1) We have other results for the theory of pointwise multipliers on Campanato spaces.(2) On several operators used in the theory of partial differential equations, for example Kohn Laplacian, some results on the normal space of homogeneous type are adaptable to general spaces of homogeneous type.(3) On the fractional integral and derivative, some results on the normal space of homogeneous type are adaptable to general spaces of homogeneous type.