Boundedness and compactness of the two-dimensional rectangular Hardy operator

Criteria in terms of weight functions $v$ and $w$ on $\mathbb{R}^2_+$ are obtained for the two-dimensional rectangular integration operator to be bounded and compact from a weighted Lebesgue space $L^p_v(\mathbb{R}^2_+)$ to $L^q_w(\mathbb{R}^2_+)$ when 1 $<p$, $q<\infty$. For $p<q$, the boundedness criterion significantly strengthens the classical result of E. Sawyer (see the Introduction) for $p\le q$. The case $q<p$ is also discussed.