Weighted inequalities for the Sawyer two-dimensional Hardy operator and its limiting geometric mean operator

Abstract

We consider and a corresponding geometric mean operator. E. T. Sawyer showed that the Hardy-type inequality could be characterized by three independent conditions on the weights. We give a simple proof of the fact that if the weight is of product type, then in fact only one condition is needed. Moreover, by using this information and by performing a limiting procedure we can derive a weight characterization of the corresponding two-dimensional Pólya-Knopp inequality with the geometric mean operator involved.