Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Martingale transform and Square function: some weak and restricted weak
  sharp weighted estimates   Following the ideas of A. Lerner, F. Nazarov, S. Ombrosi from [12] we prove
that there is a sequence of weights $w\in A^d_1$ such that $[w]^d_{A_1}\to
\infty$, and martingale transforms $T$ such that with an absolute positive $c$
  $\|T: L^1(w) \to L^{1, \infty}(w)\| \ge c [w]^d_{A_1}\log [w]^d_{A_1}$.
  We also show the existence of the sequence of weights (now in $A_2$) such
that $[w]^d_{A_2}\to \infty$, and such that the following holds:
  $[w]_{A_2^d}\asymp \|M^d\|_{w^{-1}}^2$;
  $\|S_{w}: L^{2} (w) \to L^2(w^{-1})\| \ge c\, \|M^d\|_{w^{-1}}\sqrt{\log
\|M^d\|_{w^{-1}}}$;
  $\|S_{w}: L^{2,1} (w) \to L^2(w^{-1})\| \ge c\, \|M^d\|_{w^{-1}}\sqrt{\log
\|M^d\|_{w^{-1}}}; $
  $\|S: L^2(w)\to L^{2, \infty}(w)\|=\|S_{w^{-1}}: L^{2}(w^{-1}) \to
L^{2,\infty}(w)\|\le C\, \|M^d\|_{w^{-1}}\le C\, ([w]^d_{A_2})^{1/2}$. Finally,
it is shown that for test functions of the form $\chi_I$ the weak norm $\|S_w
\chi_I\|_{L^{2, \infty}} \le C\, [w]_{A_2^d}^{1/2}\|\chi_I\|_w$.

34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas