Boundedness of integral operators on decreasing functions

We continue the study of the boundedness of the operator S(a)f(t) = integral(infinity)(0) a(s)f(st)ds on the set of decreasing functions in L-p(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak- type estimate S-a : L-dec(p)(w) -> L-p,L-infinity(w) was completely solved, the characterization of the weights w such that S-a : L-dec(p)(w) -> L-p(w) is bounded is still open for the case in which p > 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.