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.
Categories: math.CA