Journal of Inequalities and Applications
Volume 2005 (2005), Issue 4, Pages 387-394

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

Anna Wedestig

Department of Mathematics, Luleå University, Luleå 97 187, Sweden

Received 3 November 2003

Copyright © 2005 Anna Wedestig. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider Tf=0x10x2f(t1,t2)dt1dt2 and a corresponding geometric mean operator Gf=exp(1/x1x2)0x10x2logf(t1,t2)dt1dt2. E. T. Sawyer showed that the Hardy-type inequality TfLuqCfLvp could be characterized by three independent conditions on the weights. We give a simple proof of the fact that if the weight v 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 G involved.