proceedings of the american mathematical society Volume 120, Number 3, March 1994 WEIGHTED INEQUALITIESFOR GEOMETRIC MEANS B. OPIC AND P. GURKA (Communicated by Andrew M. Bruckner) Abstract. A characterization of weights u, v is given for which the geometric mean operator Tf{x) = exp(i ^\nf{t)dt), defined for / positive a.e. on (0, co), is bounded from LP{{Q,oo); vdx) to L«((0, oo); udx),p e (0, oo) and q e [p, oo). 1. Introduction We investigate the inequality (1.1) ( r expQ l*lnf(t)dt\ u(x)dx\ < C ( H f(x)v(x)dx\ ' where / is positive a.e. on (0, oo), u, v are weights, p e (0, oo), q e [p, oo), and C is a positive constant independent of /. If p = q = 1, then (1.1) is the limit case of the well-known Hardy inequality involving the averaging operator £ J0X f(t) dt, and its characterization was given by Heinig, Kerman, and Krbec (cf. [3]). If, in addition, u(x) = I =v(x), then (1.1) holds with C = e (this result is due to Knopp, see, e.g., [2, Theorem 335]). One can show that an easy modification of ideas used in [3] provides the following result which generalizes that of [3] (for notation see below). Theorem. Let p e (0, oo), q e [p, oo), and let u,ve W(0, oo). Then there exists C e (0, oo) such that the inequality (1.1) holds for all f e LP+ ((0, oo); v) if and only if there is a > 1 such that (1.2) 5(a,,,,):=supx(-)/^/;^[exp(l|ln^^)]?^y9<oo. Moreover, if C is the least constant for which (1.1) holds, then (1.3) supf^—l-] B(a,q,p)<C <infe{a-l)/p B(a,q,p). a>l V a J a>x Received by the editors March 30, 1992 and, in revised form, June 15, 1992. 1991 Mathematics Subject Classification. Primary 26D15, 26D10; Secondary 47H19. ©1994 American Mathematical Society 771 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use