On weighted Hardy inequality with two-dimensional rectangular operator -- extension of the E. Sawyer theorem

A characterization is obtained for those pairs of weights $v$ and $w$ on $\mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(\mathbb{R}^2_+)$ to $L^q_w(\mathbb{R}^2_+)$ for $1<p\not= q<\infty$, which is an essential complement to E. Sawyer's result \cite{Saw1} given for $1<p\leq q<\infty$. Besides, we declare that the E. Sawyer theorem is actual if $p=q$ only, for $p<q$ the criterion is less complicated. The case $q<p$ is new.