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

We consider Tf= 0 x1 0 x2 f ( t1, t2) d t1 d t2 and a corresponding geometric mean operator Gf=exp ( 1/ x1 x2) 0 x1 0 x2 logf ( t1, t2) d t1 d t2 . E. T. Sawyer showed that theHardy-type inequality Tf Luq C f Lvp could be characterized by three independentconditions on the weights. We give a simple proof of the fact thatif the weight v is of product type, then in fact only onecondition is needed. Moreover, by using this information and byperforming a limiting procedure we can derive a weightcharacterization of the corresponding two-dimensional Pólya-Knopp inequality with the geometric mean operator G involved.Validerad; 2005; 20070116 (evan);Full text license: CC BY