Weighted inequalities for the one-sided geometric maximal operators

We characterize the pairs of weights (u, v) such that the one-sided geometric maximal operator G(+), defined for functions f of one real variable by G(+) f(x) = sub(h>0) exp (1/h integral(x+h)(x) log vertical bar f vertical bar), verifies the weak-type inequality integral({x is an element of R:G+ f(x) > lambda}) u <= C/lambda(p) integral(infinity)(0) vertical bar f vertical bar(p)v or the strong type inequality O(R)(G(+) f)(p) u <= c integral(R) vertical bar f vertical bar(p) v for 0 < p < infinity. We also find two new conditions which are equivalent to A(infinity)(+). (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim