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

Abstract

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 $L_v^p({\mathbb{R}}_{+}^2)$ to $L_w^q({\mathbb{R}}_{+}^2)$ for $1<p\ne q<\infty$, which is an essential complement to E. Sawyer's result \cite{Saw1} given for $1<p\le q<\infty$. 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.