Boundedness of Commutators on Hardy Spaces and Herz-type Hardy Spaces

let  be commutators generated by Lipsch itz functions b∈Lipβ(Rn) and a linear operator T with a kernel b eing of some smooth conditions. We studied the boundedne ss of the c ommutators on Hardy spaces and Herz-type Hardy spaces by using their atom characterization. The results show that if nn+βp≤1 and 1q=1p-βn, then  is bounded from Hp(Rn) to Lq(Rn); and if α,q,q 1,q2 satisfy some conditions, then  is bounded from Hα,pq1(Rn) to α,pq2(Rn). 