JournalszaaVol. 16, No. 4pp. 801–829

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

  • Y. Rakotondratsimba

    Institut Polytechnique St. Louis, Cergy-Pontoise, France
Weighted Norm Inequalities for Riemann-Liouville Fractional Integrals of Order Less than One cover
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Abstract

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