Hardy type operators on grand Lebesgue spaces for non-increasing functions

We characterize the weights ww for which the operator Tψf(x)=∫0xψ(x,y)f(y)dy is bounded between weighted grand Lebesgue spaces Lwp) for non-increasing functions. The conjugate of TψTψ, for a special ψψ, given by Sϕ∗f(x):=∫x∞f(y)ϕ(y)Φ(y)dy is considered. An extrapolation type result giving Lp)Lp)-boundedness of Sϕ∗ for non-increasing functions has been proved. Also its LpLp-boundedness has been characterized. Finally, a variant of Sϕ∗ has been considered and discussed.