A simpler proof of an inequality of Muckenhoupt and Wheeden is given. Let T α f ( x ) = ∫ f ( y ) | x − y | α − d d y {T_\alpha }f(x) = \smallint f(y)|x - y{|^{\alpha - d}}dy be given for functions defined in R d {{\mathbf {R}}^d} . Let υ \upsilon be a weight function which satisfies \[ ( | Q | − 1 ∫ Q [ υ ( x ) ] q d x ) 1 / q ( | Q | − 1 ∫ Q [ υ ( x ) ] − p ′ d x ) 1 / p ′ ≤ K (|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^q}dx{)^{1/q}}(|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^{ - p’}}dx{)^{1/p’}} \leq K} } \] for each cube, Q Q , with sides parallel to a standard system of axes and | Q | |Q| is the measure of such a cube. Suppose 1 / q = 1 / p − α / d 1/q = 1/p - \alpha /d and 0 > α > d , 1 > p > d / α 0 > \alpha > d,1 > p > d/\alpha . Then there exists a constant such that | | ( T α f ) υ | | q ≤ C | | f υ | | p ||({T_\alpha }f)\upsilon |{|_q} \leq C||f\upsilon |{|_p} . Certain results for p = 1 p = 1 and q = ∞ q = \infty are also given.