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

We characterize the weights w for which the operator T(psi)f (x) = integral(x)(0)psi(x, y)f(y)dy is bounded between weighted grand Lebesgue spaces L-w(p)) for non-increasing functions. The conjugate of T-psi, for a special psi, given by S(phi)(*)f(x) := integral(infinity)(x) f(y)phi(y)/phi(y)dy is considered. An extrapolation type result giving L-p) -boundedness of S-phi(*) for non-increasing functions has been proved. Also its L-p -boundedness has been characterized. Finally, a variant of S-phi(*) has been considered and discussed. (C) 2016 Ivane Javakhishvili Tbilisi State University. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license.