The Weighted Estimates For The Maximal Commutator Of Singular Integral Operator And Commutator Of Fractional Integral Operator On Spaces Of Homogeneous Type

The triple(χ,d,μ) is said to be a space of homogeneous type,ifχis a set endowed with a quasi-metric d and a positive Borel regular measureμ.Moreover,μsatisfies the following doubling condition: there exists a constant C≥1 such that for all x∈χand r＞O,μ(B(x,2r))≤C_μ(B(x,r))＜∞, where B(x,r) = {y∈χd(x,y)＜r} is the ball centered at x and having radius r.This dissertation is devoted to the study of the behaviours of two kinds of commutators.A weighted L~p estimate for the maximal commutator of singular integral operator is obtained.It is also given a weighted weak type norm inequality for commutator of fractional integral operator,then there is the application of this estimate.It consists of two chapters. The first chapter deals with L~p(χ)-boundedness with general weights for the maximal operator associated with the commutator generated by singular integral operator and BMO function on spaces of homogeneous type.The second chapter is concerning with an endpoint estimate with general weights for the operator associated with the commutator generated by fractional integral operator and BMO function on spaces of homogeneous type.As an application,a two-weight,weak type norm inequality for this commutator is established.