Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Variation of Calder\'on--Zygmund Operators with Matrix Weight   Let $p\in(1,\infty)$, $\rho\in (2, \infty)$ and $W$ be a matrix $A_p$ weight.
In this article, we introduce a version of variation
$\mathcal{V}_{\rho}({\mathcal T_n}_{\,,\,\ast})$ for matrix Calder\'on--Zygmund
operators with modulus of continuity satisfying the Dini condition. We then
obtain the $L^p(W)$-boundedness of $\mathcal{V}_{\rho}({\mathcal
T_n}_{\,,\,\ast})$ with norm \begin{align*} \|\mathcal{V}_{\rho}({\mathcal
T_n}_{\,,\,\ast})\|_{L^p(W)\to L^p(W)}\leq C[W]_{A_p}^{1+{1\over p-1} -{1\over
p}} \end{align*} by first proving a sparse domination of the variation of the
scalar Calder\'on--Zygmund operator, and then providing a convex body sparse
domination of the variation of the matrix Calder\'on--Zygmund operator. The key
step here is a weak type estimate of a local grand maximal truncated operator
with respect to the scalar Calder\'on--Zygmund operator.

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The categories of this paper are: comp-gas
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The categories of this paper are: econ
10
The categories of this paper are: q-bio
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The categories of this paper are: adap-org
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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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: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-th