| In this paper,we discuss the boundedness and endpoint estimates of the Hardy operator P,its adjoint operator Q and their commutators generated by P and Q with CMO functions on the weighted central Morrey spaces,the boundedness and endpoint estimates of the fractional Hardy operator Pα,its adjoint operator Qα and their commutators generated by Pa and Qa with CMO functions on the weighted central Morrey spaces.For the boundedness of the Hardy operator P and its adjoint operator Q on the weighted central Morrey spaces,we gain the results as follows.(1)Let 1<p<∞,0<κ<1,w∈Ap,0,σ=w-p’/p.Then P is bounded on L0p,κ(w).(2)Let 1<p<∞,w∈Ap,0,0<κ<βw/θw.Then Q is bounded on L0p,κ(w).For the endpoint estimates of the Hardy operator P and its adjoint operator Q on the weighted central Morrey spaces,we gain the results as follows.(1)Let p=1,w∈A1,0,0<κ<1.Then P is bounded from L01,κ(w)to WL01,κ(w).(2)Let p=1,w∈A1,0,0<κ<βw/θw.Then Q is bounded on L01,κ(w).For the boundedness of the commutators generated by P and Q with CMO functions on the weighted central Morrey spaces,we gain the results as follows.(1)Let 1<p<∞.If there is r>1 such that wr∈Ap,0.0<κ<1,bκCMOr’ max{p,p’},σ=w-p’/p.Then Pb is bounded on L0p,k(w).(2)Let 1<p<∞.If there is r>1,such that wr∈Ap,0,0<κ<min{1/θwr’+βwr/θwr},b∈CMOr’max{p,p’}.Then Qb is bounded on L0p,k(w).In addition,we also study the boundedness of the fractional Hardy operator Pα,its adjoint operator Qa and their commutators on the weighted central Morrey spaces,and gain the similar results. |
