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Zeitschrift für Analysis und ihre Anwendungen


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Volume 16, Issue 4, 1997, pp. 801–829
DOI: 10.4171/ZAA/793

Published online: 1997-12-31

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

Y. Rakotondratsimba[1]

(1) Institut Polytechnique St. Louis, Cergy-Pontoise, France

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.

Keywords: Weighted inequalities, Riemann-Liouville operators, convolution operators

Rakotondratsimba Y.: Weighted Norm Inequalities for Riemann-Liouville Fractional Integrals of Order Less than One. Z. Anal. Anwend. 16 (1997), 801-829. doi: 10.4171/ZAA/793