The Boundedness Of Commutator Of Vector-Valued Singular Integral Operators

This dissertation discussed the boundedness of commutators of vector-valued singular integral operators generated by Lipschitz function.Let 1<s<∞,f={f1,f2,…,fn,…} where fj(j=1,2,3…)is a s-mooth functions with compact supported set,define |f(x)|s=(∑j=1 ∞ |fj(x)|s)1/s,T is a generalized Calderon-Zygmund operator,define a vector-valued singular integral operator associated with |Tf(x)|s=(∑j=1∞|Tfj(x)|s)1/s,and a commuta-tor generated by T and Lipschitz function,|[b,T]f|s=(∑j=1 ∞|[b,T]fj|s)1/s,where[b,T]fj=b(x)Tfj(x)-T(bfj)(x).Set b ∈ Lipβ,0<β≤1,1<p<q<g<∞,β=n(1/p-1/q),2g’≤s,then:(1)|[b,T]f|s is bounded from Lp(Rn)to Lq(Rn).(2)|[b,T]f|s is bounded from Lp(Rn)to Fpβ,∞(Rn).