Weighted norm inequalities for fractional integrals

The principal problem considered is the determination of all nonnegative functions, V(x), such that IIhTf(x)V(x)14 < ClIf(x)V(x)lIp where the functions are defined on R', 0 < y < n, 1 < p < n/y, l/q = l/p -yn, C is a constant independent of f and Tyf (x) = S f(x y) Ly|f dy. The main result is that V(x) is such a function if and only if (-fQ [V(X)]qdx (jff [vx) P'd)1 (IQIJQVX]^ (QI J [V(X)]-p, ) where Q is any n dimensional cube, IQI 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 q = oo and a weighted version of the Sobolev imbedding theorem are also proved.