Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two-weight $L^p$-$L^q$ bounds for positive dyadic operators: unified
  approach to $p\leq q$ and $p>q$   We characterize the $L^p(\sigma)\to L^q(\omega)$ boundedness of positive
dyadic operators of the form $
  T(f\sigma)=\sum_{Q\in\mathscr{D}}\lambda_Q\int_Q f\,\mathrm{d}\sigma\cdot
1_Q, $ and the $L^{p_1}(\sigma_1)\times L^{p_2}(\sigma_2)\to L^q(\omega)$
boundedness of their bilinear analogues, for arbitrary locally finite measures
$\sigma,\sigma_1,\sigma_2,\omega$. In the linear case, we unify the existing
"Sawyer testing" (for $p\leq q$) and "Wolff potential" (for $p>q$)
characterizations into a new "sequential testing" characterization valid in all
cases. We extend these ideas to the bilinear case, obtaining both sequential
testing and potential type characterizations for the bilinear operator and all
$p_1,p_2,q\in(1,\infty)$. Our characterization covers the previously unknown
case $q<\frac{p_1p_2}{p_1+p_2}$, where we introduce a new two-measure Wolff
potential.

15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg