Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quantitative weighted bounds for the $q$-variation of singular integrals
  with rough kernels   In this paper, we study the quantitative weighted bounds for the
$q$-variational singular integral operators with rough kernels. The main result
is for the sharp truncated singular integrals itself
  $$ \|V_q\{T_{\Omega,\varepsilon}\}_{\varepsilon>0}\|_{L^p(w)\rightarrow
L^p(w)}\leq c_{p,q,n} \|\Omega\|_{ L^\infty}(w)_{A_p}^{1+1/q}\{w\}_{A_p},$$
where the quantity $(w)_{A_p}$, $\{w\}_{A_p}$ will be recalled in the
introduction; we do not know whether this is sharp, but it is the best known
quantitative result for this class of operators, since when $q=\infty$, it
coincides with the best known quantitative bounds by Di Pilino--Hyt\"{o}nen--Li
or Lerner. In the course of establishing the above estimate, we obtain several
quantitative weighted bounds which are of independent interest. We hereby
highlight two of them. The first one is
  $$ \|V_q\{\phi_k\ast T_{\Omega}\}_{k\in\mathbb Z}\|_{L^p(w)\rightarrow
L^p(w)}\leq c_{p,q,n} \|\Omega\|_{ L^\infty}(w)_{A_p}^{1+1/q}\{w\}_{A_p},$$
where $\phi_k(x)=\frac1{2^{kn}}\phi(\frac x{2^k})$ with $\phi\in
C^\infty_c(\mathbb R^n)$ being any non-negative radial function, and the
sharpness for $q=\infty$ is due to Lerner; the second one is
  $$
\|\mathcal{S}_q\{T_{\Omega,\varepsilon}\}_{\varepsilon>0}\|_{L^p(w)\rightarrow
L^p(w)}\leq c_{p,q,n} \|\Omega\|_{ L^\infty}(w)_{A_p}^{1/q}\{w\}_{A_p},$$ and
the sharpness for $q=\infty$ follows from the Hardy--Littlewood maximal
function.

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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: funct-an