grand maximal operator $\mathcal{M}_T(f)(x) = \sup_{Q\ni x}{\operatornamewithlimits{ess\,sup}}_{\xi\in Q}{\big|T(f1_{{\mathbb{R}}^d\setminus3Q})(\xi)\big|_{}},$

iii) and its localized version $\mathcal{M}_{T,Q}(f)(x) = \sup_{Q\supset P\ni x}{\operatornamewithlimits{ess\,sup}}_{\xi\in P}{\big|T(f1_{3Q\setminus3P})(\xi)\big|_{}},$ where $Q,P$ are cubes.

The control over the grand maximal operator is given by

[@L Lemma 3.2]\[keylemma\] Let $f\in L^2_{loc}.$ The following pointwise estimates hold:

i)  for a.e. $x\in Q$ we have: ${\big|T(f1_{3Q}(x))\big|_{}} \leq C_d{\big\|T\big\|_{L^1\to L^{1,\infty}}}{|f(x)|}+ \mathcal{M}_{T,Q}f(x),$

ii) for all $x\in{\mathbb{R}}^d$ we have: $\mathcal{M}_Tf(x) \leq C_d({\big\|\omega\big\|_{{\rm{Dini}}}}+C_T) \mathcal{M}f(x)+T_*f(x).$

For a more refined argument for the sparse domination in Theorem \[sparsedom\] without Lemma \[keylemma\], see the latest version of the sparse domination principle in Lerner, Ombrosi [@LO19].

\[sparsedom\] Let $T$ be a d-dimensional Calderón-Zygmund operator with a Dini kernel and denote $b(x,y) = (b_1(x)-b_1(y)) (b_2(x)-b_2(y).$ We assume that $f\in L^1_c({\mathbb{R}}^d),$ and further to make everything well-defined that $b_1,b_2,b_1b_2,b_1f,b_2f,b_1b_2f\in L^1_{loc}.$

From these assumptions it follows that there exists a sparse collection $S$ of cubes on ${\mathbb{R}}^d$ such that $${\Big|C_bTf(x)\Big|_{}} \leq C_{T,d} \sum_{i=1}^{4}S_if(x),$$ where $$S_1f = \sum_{Q\in S} {|b_1-\langle b_1\rangle_Q|} {|b_2-\langle b_2\rangle_Q|}\big\langle{|f|}\big\rangle_Q1_Q, \qquad
    S_2f = \sum_{Q\in S} {|b_2-\langle b_2\rangle_Q|}\big\langle{|b_1-\langle b_1\rangle_Q|}{|f|}\big\rangle_Q1_Q,$$ $$S_3f = \sum_{Q\in S} {|b_1-\langle b_1\rangle_Q|}\big\langle{|b_2-\langle b_2\rangle_Q|}{|f|}\big\rangle_Q1_Q,\qquad
    S_4f = \sum_{Q\in S} \big\langle{|b_1-\langle b_1\rangle_Q|}{|b_2-\langle b_2\rangle_Q|}{|f|}\big\rangle_Q1_Q,$$ and the sparse constant denoted with $\gamma$ depends only on the dimension $d.$

We recall only the part of the proof 