Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin