Communications in Contemporary Mathematics ( IF 1.278 ) Pub Date : 2020-10-24 , DOI: 10.1142/s0219199720500625
Xuan Thinh Duong; Ji Li; Dongyong Yang

Let $p∈(1,∞)$, $ρ∈(2,∞)$ and $W$ be a matrix $Ap$ weight. In this paper, we introduce a version of variation $𝒱ρ(𝒯n,∗)$ for matrix Calderón–Zygmund operators with modulus of continuity satisfying the Dini condition. We then obtain the $Lp(W)$-boundedness of $𝒱ρ(𝒯n,∗)$ with norm $∥𝒱ρ(𝒯n,∗)∥Lp(W)→Lp(W)≤C[W]Ap1+1p−1−1p$ 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.

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