Weighted norm inequalities for fractional integrals

Abstract:

The principal problem considered is the determination of all nonnegative functions, $V(x)$, such that $\left \|{T_\gamma }f(x)V(x)\right \|_q \leq C\left \|f(x)V(x)\right \|_p$ 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 ${T_\gamma }f(x) = \smallint f(x - y)|y{|^{\gamma - n}}dy$. The main result is that $V(x)$ is such a function if and only if \[ {\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^q}dx} } \right )^{1/q}}{\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^{ - p’}}dx} } \right )^{1/p’}} \leq K\] 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 $q = \infty$ and a weighted version of the Sobolev imbedding theorem are also proved.