JournalszaaVol. 15, No. 1pp. 75–93

Weighted Inequalities for the Fractional Integral Operators on Monotone Functions

  • Y. Rakotondratsimba

    Institut Polytechnique St. Louis, Cergy-Pontoise, France
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Abstract

We give a characterization of weight functions uu and vv on Rn\mathbb R^n for which the fractional integral operator lsl_s of order ss on Rn\mathbb R^n defined by (Isf)(x)=Rnxysnf(y)dy(I_s f)(x) = \int_{\mathbb R^n} | x - y |^{s–n} f(y)dy sends all monotone functions which belong to the weighted Lebesgue space Lvp(Rn)L^p_v(\mathbb R^n) into the weighted Lebesgue space Luq(Rn)L^q_u(\mathbb R^n). This characterization is done for all pp and qq with 1<p<1 < p < \infty and 0<q<0 < q < \infty. The analogous Lorentz and Orlicz problems are also considered.

Cite this article

Y. Rakotondratsimba, Weighted Inequalities for the Fractional Integral Operators on Monotone Functions. Z. Anal. Anwend. 15 (1996), no. 1, pp. 75–93

DOI 10.4171/ZAA/689