Differential Operators Associated With Fractional Integral Operators Commutators Boundedness

In this paper, firstly, the fractional integral operators L-α/2associated with differential operator L is studied, the Triebel-Lizorkin space and several results of them are introduced. By the methods similar to Plauszynski theory, we ob-tain that the commutators [b, L-α/2] with Lipschitz funtions b is bounded on the Triebel-Lizorkin space. Secondly, we recall the Morrey-Herz space and ba-sic theories of them. With the methods of ring decomposition of functions, the boundedness of commutators [b, L-α/2]with b belonging to BMO(Rn) functions on the homogeneous Morrey-Herz space is obtained. At last, the homogeneous weighted Herz space and some basic theories of them have been introduced, the properties of weight functions has been described. By the atom decompo-sitions theory on the homogeneous weighted Herz space and the property of weight functions, the boundedness of commutators [b, L-α/2] with b belonging to BMO(Rn) functions on the homogeneous weighted Herz space is obtained.