Variation of Calderón--Zygmund Operators with Matrix Weight

Abstract

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 Calderón--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}_ρ({\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 Calderón--Zygmund operator, and then providing a convex body sparse domination of the variation of the matrix Calderón--Zygmund operator. The key step here is a weak type estimate of a local grand maximal truncated operator with respect to the scalar Calderón--Zygmund operator.