Weighted Hardy inequality with two-dimensional rectangular operator: the case q<p

A criterion is obtained for the boundedness of the two-dimensional rectangular integration operator from a weighted Lebesgue space L p v (R 2 + ) to L q w (R 2 + ) for 1 < q < p < , which is a supplement to E. Sawyer's theorem [8] and its extension [9] given for 1 < p q < .