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

### Y. Rakotondratsimba

Institut Polytechnique St. Louis, Cergy-Pontoise, France

## Abstract

Necessary and sufficient condition on weight functions $u(⋅)$ and $v()$ are derived in order that the Riemann-Liouville fractional integral operator $R_{α}(0<α<1)$ is bounded from the weighted Lebesgue spaces $L_{p}((0,∞),v(x)dx)$ into $L_{q}((0,∞),u(x)dx)$ whenever $1<p≤q<∞$ or $1<q<p<∞$. 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 $(–∞,∞)$, are also solved.

## Cite this article

Y. Rakotondratsimba, Weighted Norm Inequalities for Riemann-Liouville Fractional Integrals of Order Less than One. Z. Anal. Anwend. 16 (1997), no. 4, pp. 801–829

DOI 10.4171/ZAA/793