Journal of Inequalities and Applications
Volume 2005 (2005), Issue 4, Pages 387-394
doi:10.1155/JIA.2005.387
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.