The Study On Boundedness For Multilinear Commutator Of Fractional Integral Operator On Spaces Of Homogeneous Type

In this paper, we study the boundedness for the multilinear commutator ofthe fractional integral operator on the spaces of homogeneous type., i.e., we studysystem the boundedness on Lp(1 < p <∞) spaces,Hardy spaces,Herz-Hardyspaces,Triebel-Lizorkin spaces of the multilinear commutators I_γ~b generated by thefractional integral operator Iγand BMO functions or Lipschitz functions on the spacesof homogeneous type X. Moreover, we consider some kind of endpoint estimates forthese multilinear commutators.At first, the (Lp(X),Lq(X))-boundedness for the multilinear commutator I_γ~b isproved. In this section, we adopt two methods in proving it. On the one hand, theSharp function inequality is proved, by using it, we obtain I_γ~b is bounded from Lp(X)to Lq(X), where 1 < p < 1/γand 1/q = 1/p ?γ; on the other hand, the Good-λinequality for the multilinear commutator I_γ~b is proved. By using it, we also obtain Iγbis bounded from Lp(X) to Lq(X).Secondly, the BMO estimates for the multilinear commutator I_γ~b related to thefractional integral operator are proved. In this chapter, we contain two parts. Forone thing, we establishλ-central BMO estimates for I_γ~b on central Morrey spaces,and so does the multilinear commutator of the fractional maximal operator Mγb. Foranother thing, we prove CBMO estimates for the multilinear commutator on Herzand Morrey-Herz spaces.Thirdly, the boundedness for the multilinear commutator Iγb on Hbp (X) and H K˙qα,,bp(X) are proved, where bi∈BMO(X),1≤i≤m, b = (b1,···,bm).Then, we get that Iγb is bounded from Lp(X) to F˙pmβ,∞(X), where 1 < p < 1/γand 1/p ? 1/q =γ; Lp(X) to Lq(X), where 1/p ? 1/q = mβ+γand 1/p > mβ+γ;Hp(X) to Lq(X); H K˙qα1, p(X) to K˙qα2, p(X); H K˙q11? 1/q1+γ,p(X) to W K˙q12? 1/q1+γ,p(X),where Iγb is generated by the fractional integral operator and functions in Lipschitzspace.Finally, the endpoint estimates for the multilinear commutator Iγb are studied.They are bounded from L1/γto BMO(X). Moreover, let 0 <γ< 1, 1 < p < 1/γandb = (b1,···,bm) with bj∈BMO(X) for 1≤j≤m. Then Iγb is bounded from Bpγ(X)to CMO(X). Last, if for any H1(X)-atom a supported on certain ball B and u∈B,there is(?) then Iγb is bounded from H1(X) to L1/(1?γ)(X).