Weighted Inequalities of Weak Type for the Fractional Integral Operator

Abstract

Sufficient conditions on weights $u(\cdot)$ and $v(\cdot)$ are given in order that the usual fractional integral operator $I_{\alpha} (0 < \alpha < n)$ is bounded from the weighted Lebesgue space $L^p(v(x)dx)$ into weak-$L^p(u(x)dx)$, with $1 ≤ p < \infty$. As a consequence a characterization for this boundedness is obtained for a large class of weight functions which particularly contains radial monotone weights.