The Boundedness Of Hardy Operator And Commutators On Weighted Herz Spaces

In this paper,we discuss the boundedness and the endpoint estimates of the Hardy operator P,its adjoint operator Q and their commutators generated by P and Q with CMO function on the weighted Herz spaces,the boundedness and the endpoint estimates of the fractional Hardy operator Pβ,its adjoint operator Qβ and their commutators generated by Pβ and Qβ with CMO function on the weighted Herz spaces.For the boundedness of the Hardy operator P and its adjoint operator Q on the weighted Herz spaces,we gain the results as follows.(1)Let 1<q<∞,0<p<∞,w∈Aq,0 and α<γσ/θwq’ where σ=w-q’/q.Then P is bounded on Kqα,p(w).(2)Let 1<q<∞,0<p<∞,w∈Aq,0 and α>-1/q.Then Q is bounded on Kqα,p(w).For the endpoint estimates of the the Hardy operator P and its adjoint operator Q on the weighted Herz spaces,we gain the results as follows.(1)Let q=1,0<p<∞,w∈A1,0 and α<0.Then P is bounded on K1α,p(w).(2)Let q=1,0<p<∞,w∈A1,0 and α>-1.Then Q is bounded on K1α,p(w).For the boundedness of the commutators generated by P and Q with CMO function on the weighted Herz spaces,we gain the results as follows.(1)Let 1<g<∞,0<p<∞.If there is r>1,wr∈Aq,0,b∈CMOr’max{q,q’} andα<1/θwr’q’+γσr/θwrq’ where σ=w-q’/q.Then Pb is bounded on Kqα,p(w).(2)Let 1<q<∞),0<p<∞.If there is r>1,wr∈Aq,0,b∈CMOr’ max{q,q’} andα>1/γwqr’-γwr/γwqr.Then Qb is bounded on Kqα,p(w).In addition,we also study the boundedness of the fractional Hardy operator Pβ,its adjoint operator Qβ and their commutators on the weighted Herz spaces,and gain the similar results.