Weighted Norm Inequalities for Fractional Integrals

The principal problem considered is the determination of all nonnegative functions, $V(x)$, such that \|T_γ f(x)V(x) \|q ≤ C \|f(x)V(x) \|ₚ where the functions are defined on Rⁿ,0 < γ < n,1 < p < n/γ ,1/q = 1/p - γ /n, C is a constant independent of f and T_γ f(x) = f(x - y)|y|γ - ndy. The main result is that $V(x)$ is such a function if and only if \[ { ( { {1}{{|Q|}}∫ _Q {{{[V(x)]}^q}dx} } )1/q}{ ( { {1}{{|Q|}}∫ _Q {{{[V(x)]}- p’}dx} } )1/p’} ≤ K\] where Q is any n dimensional cube, $|Q|$ 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 = ∞ and a weighted version of the Sobolev imbedding theorem are also proved.