Journal of Inequalities and Applications
Volume 2005 (2005), Issue 4, Pages 387-394
doi:10.1155/JIA.2005.387
Weighted inequalities for the Sawyer two-dimensional Hardy operator and its
limiting geometric mean operator
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.
Abstract
We consider
Tf=∫0x1∫0x2f(t1,t2)dt1dt2
and a corresponding geometric mean operator Gf=exp(1/x1x2)∫0x1∫0x2logf(t1,t2)dt1dt2. E. T. Sawyer showed that the
Hardy-type inequality ‖Tf‖Luq≤C‖f‖Lvp 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.