# Weighted Inequalities for the Fractional Maximal Operator and the Fractional Integral Operator

### Y. Rakotondratsimba

Institut Polytechnique St. Louis, Cergy-Pontoise, France

## Abstract

A sufficient condition is given on weight functions $u$ and $v$ on $R_{n}$ for which the fractional maximal operator $M_{s}(0≤s<n)$ defined by $(M_{s}f)(x)=sup_{Q∋x}∣Q∣_{ns–1}∫_{Q}∣f(y)∣dy$ or the fractional integral operator $I_{s}(0<s<n)$ defined by $(I_{s},f)(x)=∫_{R_{n}}∣x−y∣_{s–n}f(y)dy$ is bounded from $L_{p}(R_{n},vdx)$ into $L_{q}(R_{n},udx)$ for $0<q<p$ with $p>1$, where $Q$ is a cube and n a non-negative integer.

## Cite this article

Y. Rakotondratsimba, Weighted Inequalities for the Fractional Maximal Operator and the Fractional Integral Operator. Z. Anal. Anwend. 15 (1996), no. 2, pp. 309–328

DOI 10.4171/ZAA/702