proceedings of the american mathematical society Volume 118, Number 1, May 1993 TWO WEIGHT ^-INEQUALITIES FOR THE HARDY OPERATOR, HARDY-LITTLEWOOD MAXIMAL OPERATOR, AND FRACTIONAL INTEGRALS LAI QINSHENG (Communicated by J. Marshall Ash) Abstract. Suppose <J> is an appropriate Young's function and w(x), v{x) are nonnegative locally integrable functions. Let T denote one of three lin- ear operators of special importance that map suitable functions on R" into functions on R" . For the Hardy operator T, we study the inequality roo roo / <t>(\Tf(x)\)w(x)dx<C <b(\f(x)\)v(x)dx Jo Jo and for the Hardy-Littlewood maximal operator or fractional integrals T, we discuss the inequalities j <t>(\T(fv)(x)\)w(x)dx<C j 4>(\f(x)\)v(x)dx. Jr" Jr" In all cases we obtain the necessary and sufficient conditions. 1. Introduction We shall be concerned with integral inequalities of the form (1.1) / ®\Tf(x)\)dw<C f <D(|/(x)|)c7/7, JR" JR" where dw, dp are positive Borel measures on R", <P is an even Young's function on 7? with 0(0) = 0, and T is one of three linear operators of special importance that map suitable functions on 7?" into functions on 7?" . Apart from their intrinsic interest, such inequalities are important in application, since they imply the boundedness of T as a map between the associated Orlicz spaces. It is of particular interest to obtain estimates for the best constant C in (1.1). The first of the cases we consider is the Hardy operator (1.2) Tf(x)= f f(t)dt, xe7?+ = (0,co). Jo Received by the editors September 24, 1990 and, in revised form, August 19, 1991. 1991 Mathematics Subject Classification. Primary 42B25. Key words and phrases. Young's function, Hardy operator, maximal operator, fractional integral. This work was supported in part by the Universities' China Committee in London. ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 129