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

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

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20wm01snus 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 sennheiser ba 70 performing a limiting procedure we can derive a weight characterization of the corresponding two-dimensional Pólya-Knopp inequality with the geometric mean operator involved.