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Weighted inequalities for the Sawyer two-dimensional Hardy operator and its limiting geometric mean operator
Journal of Inequalities and Applications volume 2005, Article number: 374053 (2005)
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.
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Wedestig, A. Weighted inequalities for the Sawyer two-dimensional Hardy operator and its limiting geometric mean operator. J Inequal Appl 2005, 374053 (2005). https://doi.org/10.1155/JIA.2005.387