The principal problem considered is the determination of all nonnegative functions, U(x), for which there is a constant, C, such that

Abstract. The principal problem considered is the determination of all nonnegative functions, U(x), for which there is a constant, C, such that

ABSTRACT. The principal problem considered is the determination of all nonnegative functions, V(x), such that ||7;/(jc)K(a:)||, < C||/(x)K(x)||, where the functions are defined on R", 0 < y < n, 1 < p < n/y, \/q = \/p -y/n, C is a constant independent of / and Tyf(x) = ff(x -yïiW'dy. The main result is that V(x) is such a function if and only if (Híírwp*r(aiirwr*rsjr where Q is any n dimensional cube, |ß| denotes the measure of Q, p' = p/(p -1) and K is a constant independent of Q. Substitute results for the cases p = 1 and 9=00 and a weighted version of the Sobolev imbedding theorem are also proved.1. Introduction. The first norm inequality for fractional integrals was the one proved by Hardy and Littlewood in [6] for the one dimensional case with V(x) = 1; they also proved a result for V(x) = \x\". The result in n dimensions with V(x) = 1 was obtained by Sobolev in [8] and with V(x) = |x|a by Stein and G. Weiss in [10]. T. Walsh in [12] obtained a result for other weight functions and with a more general operator but did not characterize all such V 's.A slightly stronger result is obtained here than stated in the abstract. It is shown that

No abstract

The principal problem considered is the determination of all nonnegative functions W(x) with period 277 such that i" \Jie)\pwie)de The main result is that W(x) is such a function if and only if 1/7-1 7(0)= iim-f fie-4>)d4> * Am.+ 77 Jes||