ON THE GEOMETRIC MEAN OPERATOR

We give a characterization of pairs of weights (u, v) such that the ge ometric mean operator Gf(x) = exp((1/x) integral-x/0 logf(t) dt), derm ed for f > 0 almost everywhere on (0, infinity), is bounded from L(p,v ) (0, infinity) to L(q,u) (0, infinity), where 0 < q < p less-than-or- equal-to infinity. Our proofs are based on the rather surprising but s imple observation that in the case v = 1 and p > 1 the good weights fo r G coincide with those good for the averaging operator Af(x) = (1/x) integral-x/0(t) dt. Our result applies to a certain independence on p, q of weighted L(p) - L(q) inequalities involving the operator A. (C) 1994 Academic Press Inc.