Weighted Norm Inequalities for Riemann-Liouville Fractional Integrals of Order Less than One

Abstract

Necessary and sufficient condition on weight functions $u(\cdot)$ and $v()$ are derived in order that the Riemann-Liouville fractional integral operator $R_{\alpha} (0 < \alpha < 1)$ is bounded from the weighted Lebesgue spaces $L^p((0, \infty),v(x)dx)$ into $L^q((0, \infty),u(x)dx)$ whenever $1 < p ≤ q < \infty$ or $1 < q < p < \infty$. As a consequence for monotone weights then a simple characterization for this boundedness is given whenever $p ≤ q$. Similar problems for convolution operators, acting on the whole real axis $(–\infty, \infty)$, are also solved.