Weighted Hardy and singular operators in Morrey spaces

We study the weighted boundedness of the Cauchy singular integral operator S Γ in Morrey spaces L p , λ ( Γ ) on curves satisfying the arc-chord condition, for a class of “radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces L p , λ ( 0 , ℓ ) , ℓ > 0 . We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy–Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.