Weighted Norm Inequalities for Fractional Integrals

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 -yiW'dy. The main result is that V(x) is such a function if and only if (Hrwp*r(aiirwr*rsjr where Q is any n dimensional cube, || 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.