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

Abstract

A sufficient condition is given on weight functions $u$ and $v$ on ${\mathbb{R}}^n$ for which the fractional maximal operator $M_s(0\le s<n)$ defined by $(M_{s}f)(x)={\sup }_{Q\ni x}|Q{|}^{\frac{s}{n}–1}{\int }_Q|f(y)|dy$ or the fractional integral operator $I_s(0<s<n)$ defined by $(I_s,f)(x)={\int }_{{\mathbb{R}}^n}|x-y{|}^{s–n}f(y)dy$ is bounded from $L^p({\mathbb{R}}^n,vdx)$ into $L^q({\mathbb{R}}^n,udx)$ for $0<q<p$ with $p>1$, where $Q$ is a cube and n a non-negative integer.