WEIGHTED CRITERIA FOR THE HARDY TRANSFORM UNDER THE B_p CONDITION IN GRAND LEBESGUE SPACES AND SOME APPLICATIONS

摘要

We show that the Hardy operator Hf(x)=1/x~x∫_0f(t)dt from L~p)_dec,w(I) to L~p_w (I), 0 < p < ∞, 6 > 0, I = (0,1), is bounded if and only if the weight w belongs to the well-known class B_p restricted to the interval I. This result is applied to derive a similar assertion for the Riemann-Liouville fractional integral operator and to establish criteria for the boundedness of the Hardy-Littlewood maximal operator in the weighted grand Lorentz space ∧~p_w,θ .