Weighted Inequalities for the Fractional Integral Operators on Monotone Functions

Abstract

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 $L_v^p({\mathbb{R}}^n)$ into the weighted Lebesgue space $L_u^q({\mathbb{R}}^n)$. 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.