Weighted Norm Inequalities for Fractional Integrals

Abstract

A simpler proof of an inequality of Muckenhoupt and Wheeden is given. Let ${T_\alpha }f(x) = \smallint f(y)|x - y{|^{\alpha - d}}dy$ be given for functions defined in ${{\mathbf {R}}^d}$. Let $\upsilon$ be a weight function which satisfies \[ (|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^q}dx{)^{1/q}}(|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^{ - p’}}dx{)^{1/p’}} \leq K} } \] for each cube, $Q$, with sides parallel to a standard system of axes and $|Q|$ is the measure of such a cube. Suppose $1/q = 1/p - \alpha /d$ and $0 < \alpha < d,1 < p < d/\alpha$. Then there exists a constant such that $||({T_\alpha }f)\upsilon |{|_q} \leq C||f\upsilon |{|_p}$. Certain results for $p = 1$ and $q = \infty$ are also given.