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Zeitschrift für Analysis und ihre Anwendungen


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Volume 15, Issue 1, 1996, pp. 75–93
DOI: 10.4171/ZAA/689

Published online: 1996-03-31

Weighted Inequalities for the Fractional Integral Operators on Monotone Functions

Y. Rakotondratsimba[1]

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

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

Keywords: Weighted inequalities, fractional integral operators, Hardy operators

Rakotondratsimba Y.: Weighted Inequalities for the Fractional Integral Operators on Monotone Functions. Z. Anal. Anwend. 15 (1996), 75-93. doi: 10.4171/ZAA/689