Suppose $T_{1},T_{2}$ are two bounded linear operators from $L^{2}(\mathbb{R^{n}})$ to $L^{2}(\mathbb{R^{n}})$. In addition, for any $f,g\in L^{2}(\mathbb{R^{n}})$ and $x\in \mathbb{R^{n}}$, $$|T_{1}(f)(x)|\leq T_{2}(|f|)(x) \quad and \quad T_{2}(|f+g|)(x)\leq T_{2}(|f|)(x)+T_{2}(|g|)(x).$$ If $T_{2}$ is a compact operator from $L^{2}(\mathbb{R^{n}})$ to $L^{2}(\mathbb{R^{n}})$, can we deduce that $T_{1}$ is a compact operator from $L^{2}(\mathbb{R^{n}})$ to $L^{2}(\mathbb{R^{n}})$?
Actually, in the article "Sparse domination results for compactness on weighted spaces", the authors used this technique to show that some Calderón-Zygmund operators have compact extensions on $L^{2}(\mathbb{R^{n}})$, but they did not explain how to get the compactness of Calderón-Zygmund operators in their article. You can find this article on https://arxiv.org/abs/1912.10290