| Hardy operator is a basic and important operator in functional theory,and have many applications in PDE,complex analysis and other mathematical fields.In this paper,we discuss the boundedness on Hardy spaces for classical Hardy operators,its adjoint operator and their commutators.Let f ba a locally integrable function on R+,the Hardy operator and its adjoint operator are defined by#12 Let b be a measurable function on R+,the commutators of P and Q with function b are defined by[b,P]f(x)=b(x)Pf(x)-P(bf)(x),[b,Q]f(x)=b(x)Qf(x)-Q(bf)(x).For the Hardy operator P and its adjoint operator Q,we obtain the following results:(1)The Hardy operators P and Q are bounded from H~1(R+)to L1(R+),more pre-cisely||Pf||H~1(R+)≤C||f||L1(R+),||Qf||H~1(R+)≤C||f||L1(R+),where C is a constant independent of f.(2)The operator Q is bounded from H~1(R+)to H~1(R-),more precisely||Qf||H~1(R-)≤C||f||H~1(R+),where C is a constant independent of f.(3)Let 1<p<∞,then P and Q are bounded from HKp(R+)to Kp(R+),more precisely||Pf||Kp(R+)≤C||f||HKp(R+),||Qf||Kp(R+)≤C||f||HKp(R+),where C is a constant independent of f.For the commutators of Hardy operator with BMO and CMO function,we obtain the following results:(1)Let b∈BMO(R+),the commutators[b,P]is bounded from H~1(R+)to L1,∞(R+),more precisely||[b,P]f||L1,∞(R+)≤C||b||BMO(R+)||f||H~1(R+),where C is a constant independent of f.(2)Let b∈BMO(R+),the commutators[b,Q]is bounded from H~1(R+)to L1(R+),more precisely||[b,Q]f||L1(R+)≤C||b||BMO(R+)||f||H~1(R+),where C is a constant independent of f.(3)Let 1<p<∞,b∈CMOp(R+),the commutators[b,Q]is bounded from Hb1(R+)to L1(R+),more precisely||[b,P]f||L1(R+)≤C||b||CMOp(R+)||f||Hb1(R+),where C is a constant independent of f.In addition,we also study the boundedness of fractional Hardy operators and their commutators on Hardy Spaces,and obtain similar results. |
