WEIGHTED INEQUALITIES FOR THE SAWYER TWO-DIMENSIONAL HARDY OPERATOR AND ITS LIMITING GEOMETRIC MEAN OPERATOR

We consider T f = ∫ x10 ∫ x20 f (t1, t2)dt1dt2 and a corresponding geometric mean operator G f = exp(1/x1x2) ∫ x1 0 ∫ x2 0 log f (t1, t2)dt1dt2. E. T. Sawyer showed that the Hardy-type in-equality ‖T f ‖Lqu ≤ C ‖ f ‖Lpv 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 corre-sponding two-dimensional Pólya-Knopp inequality with the geometric mean operator G involved. 1