Weighted Inequalities For Fractional Hardy Operator And Commutators

In this paper, we study the weighted boundedness of fractional Hardy operator and its related operator.For the maximal operator associated with the fractional Hardy operator, 0 < < 1, we obtain the following results:(1) Let 1 ≤ < 1/, 1/ = 1/- , then is bounded from () to ,∞()if and only if ∈ ,,0.(2) Let 1 < < 1/, 1/ = 1/- , then is bounded from () to () if and only if ∈ ,,0.(3) Let 1 ≤ < 1/, 1/ = 1/- , then is bounded from () to ,∞() if and only if(, ) ∈ ,.(4) Let 1 < < 1/, 1/ = 1/- , then is bounded from () to () if and only if for any > 0,(, ) satisfies(∫ 0[(-′(0,))()]())1/≤ (∫ 0()-′)1/.For two-weight inequalities for the fractional Hardy operator and its dual operator, we obtain the following results:Let 1 < < < ∞, 0 < < 1,(1)(, ) be a pair of weights for which there exists > 1 such that for every > 0,(1/+-1/)(1∫ 0())1/(1∫ 0()-′)1/′≤ ,then(∫ ∞0|)|()1/≤ (∫ ∞0|()|())1/.(2)(, ) be a pair of weights for which there exists > 1 such that for every > 0,(1/+-1/)(1∫ 0())1/(1∫ 0()-′)1/′≤ ,then(∫ ∞0|()|())1/≤ (∫ ∞0|()|())1/.For two-weight inequalities for the commutators of the fractional Hardy operator and the commutators of its dual operator , we obtain the following results:Let 1 < < < ∞, 0 < < 1, ∈ ′max {′,}, and(, ) be a pair of weights for which there exists > 1 such that for every > 0,(1/+-1/)(1∫ 0())1/(1∫ 0()-′)1/′≤ ,then(∫ ∞0|()|())1/≤ (∫ ∞0|()|())1/,and(∫ ∞0|()|())1/≤ (∫ ∞0|)|())1/.