On Sawyer Duality in One and Higher Dimensions

We develop the Sawyer duality principle for non-increasing functions where, in the denominator, the function g is replaced by ∫x∞g(t)tdt. This result complements Stepanov’s result in which g was replaced by 1x∫0xg(t)dt. We use our result and obtain Stepanov’s result for non-decreasing functions and similarly use Stepanov’s result in proving our result for non-decreasing functions. These results have also been obtained in Rn.