The Fractional Integral Operator Of The Nature Of The Swap Operator In Some Space

The fractional integral operators is also called Reisz potential. We all know that the boundedness of fractional integral operators is a very hot topic in Har-monic analysis. Meanwhile, the commutators play an important role in charac-terization of function spaces.Firstly, the classic results of the commutators in the space of Lebesgue are displayed, including the boundedness of commutators which are complished by the function of BMO function and Lipschitz function. On the other hand, the characterization of BMO function spaces and Lipschitz function spaces are con-sidered. Based on the definition of Morrey spaces and the homogenous type spaces, it isn’t difficult to conclude that Morrey spaces are more general than the spaces of Lebesgue. The homogeneous spaces are given on a quasi-metric and satisfy the doubling condition, and it is more general than Euclidean spaces. So, here the cases on Morrey spaces and homogeneous spaces are also given.