Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On an estimate of Calder\'on-Zygmund operators by dyadic positive
  operators   Given a general dyadic grid ${\mathscr{D}}$ and a sparse family of cubes
${\mathcal S}=\{Q_j^k\}\in {\mathscr{D}}$, define a dyadic positive operator
${\mathcal A}_{{\mathscr{D}},{\mathcal S}}$ by $${\mathcal
A}_{{\mathscr{D}},{\mathcal S}}f(x)=\sum_{j,k}f_{Q_j^k}\chi_{Q_j^k}(x).$$ Given
a Banach function space $X({\mathbb R}^n)$ and the maximal Calder\'on-Zygmund
operator $T_{\natural}$, we show that $$\|T_{\natural}f\|_X\le
c(n,T)\sup_{{\mathscr{D}},{\mathcal S}}\|{\mathcal A}_{{\mathscr{D}},{\mathcal
S}}f\|_{X}.$$
  This result is applied to weighted inequalities. In particular, it implies:
(i) the "two-weight conjecture" by D. Cruz-Uribe and C. P\'erez in full
generality; (ii) a simplification of the proof of the "$A_2$ conjecture"; (iii)
an extension of certain mixed $A_p$-$A_r$ estimates to general
Calder\'on-Zygmund operators; (iv) an extension of sharp $A_1$ estimates (known
for $T$) to the maximal Calder\'on-Zygmund operator $T_{\natural}$.

17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ