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 $M_s (0 ≤ s < n)$ defined by $(M_sf)(x) = \mathrm {sup}_{Q \ni x} |Q|^{\frac{s}{n}–1} \int_Q | f (y) | dy$ 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.