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

Abstract

Necessary and sufficient condition on weight functions $u(\cdot )$ and $v(\cdot )$ 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 $1<p\le q<\infty$ or $1<q<p<\infty$. As a consequence for monotone weights then a simple characterization for this boundedness is given whenever $p\le q$. Similar problems for convolution operators, acting on the whole real axis $(–\infty ,\infty )$, are also solved.