Commutator Theorems For Fractional Integrals On Stratified Lie Groups

Let G be a stratified Lie group with homogenous dimension Q and {Xj}j=1n be a basis for its left-invariant vector fields of degree one.Let L=∑j=1n Xj2 be the sub-Laplacian on G.The fractional integrals on G is defined as Iα=(-L)-α2.This thesis studies the weighted boundedness and compactness of the commutator[6,Iα]generated by Iα with a local integrable function b.The characterizations of BMOv(G)via the Bloom-type twoweighted estimates of[b,Iα]and CMO(G)via the weighted compactness of[b,Iα]are established,respectively.As an application,a factorization of H1(G),the Hardy space on G,is also obtained via the boundedness of the commutator of Iα.