On the Geometric Mean Operator

We give a characterization of pairs of weights (u, v) such that the geometric mean operator Gf(x) = exp((1/x) ∫x0 log ƒ(t) dt), defined for ƒ > 0 almost everywhere on (0, ∞), is bounded from Lp,v (0, ∞) to Lq,u (0, ∞), where 0 < q < p ≤ ∞. Our proofs are based on the rather surprising but simple observation that in the case v ≡ 1 and p > 1 the good weights for G coincide with those good for the averaging operator Af(x) = (1/x) ∫x0 ƒ(t) dt. Our result applies to a certain independence on p, q of weighted Lp − Lq inequalities involving the operator A.