Weighted norm inequalities for fractional integrals

The principal problem considered is the determination of all nonnegative functions, $V(x)$, such that $\left\Vert{T_\gamma }f(x)V(x)\right\Vert _q \leq C\left\Vert f(x)V(x)\right\Vert _p$ where the functions are defined on ${R^n},0 < \gamma < n,1 < p < n/\gamma ,1/q = 1/p - \gamma /n$, C is a constant independent of f and ${T_\gamma }f(x) = \smallint f(x - y)\vert y{\vert^{\gamma - n}}dy$. The main result is that $V(x)$ is such a function if and only if

$${\left( {\frac{1}{{\vert Q\vert}}\int_Q {{{[V(x)]}^q}dx} } \right... ...{\frac{1}{{\vert Q\vert}}\int_Q {{{[V(x)]}^{ - p'}}dx} } \right)^{1/p'}} \leq K$$

where Q is any n dimensional cube, $\vert Q\vert$ 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 $q = \infty$ and a weighted version of the Sobolev imbedding theorem are also proved.