Weighted weak type inequalities for modified Hardy operators and geometric means operators in dimensions one and greater

We characterize the pairs of weights (u,v)(u,v) such that the geometric mean operator G1G1, defined for positive functions f   on (0,∞)(0,∞) byG1f(x)=exp(1x∫0xlogf), verifies the weak type inequality(∫{x∈(0,∞):G1f(x)>λ}u)1q⩽Cλ(∫0∞fpv)1p in the case 0