\psi_i$ to attain $$\begin{split}
        &\dashint_Q{|\psi_1\psi_2|}^2 \leq {\Big|\dashint_Q\dashint_Q\left(b_1(x)-b_1(y)\right)\left(b_2(x)-b_2(y)\right) \overline{\psi_1(x)\psi_2}(x) 1_Q(y)  {\,\mathrm{d}}y{\,\mathrm{d}}x\Big|_{}} \\
        &\leq C_d\sum_{i=1}^{k}{\big\|C_bR_i\big\|_{L^p\to L^p}} \left( {\dashint_{Q} {{|\psi_1\psi_2|}^2}} \right) ^{1/2}\left( {\dashint_{Q} {{|f_2|}^{2}}} \right) ^{1/2} = C_d\sum_{i=1}^{k}{\big\|C_bR_i\big\|_{L^p\to L^p}} \left( {\dashint_{Q} {{|\psi_1\psi_2|}^2}} \right) ^{1/2}.
        \end{split}$$ Dividing out equal factors and summing gives the claim.

Sufficient conditions
=====================

Next, our focus will be on Calderón-Zygmund operators satisfying the Dini condition. We begin with partially recalling, with only minor modifications, a sparse domination from Lerner [@L] (also see Lerner, Ombrosi [@LO19]). See also Ibánez-Firnkorn - Rivera-Ríos [@IR].

\[CZO\] A $d$-dimensional Calderón-Zygmund operator $T$ with an $\omega$-Dini -kernel is a $L^2({\mathbb{R}}^d)\to L^2({\mathbb{R}}^d)$ bounded operator with the representation $$Tf(x) = \int K(x,y)f(y){\,\mathrm{d}}y, \qquad x\notin {\operatorname{spt}}(f),$$ with the kernel $K:{\mathbb{R}}^d\times{\mathbb{R}}^d\setminus\{(x,x) : x\in{\mathbb{R}}^d\}\rightarrow{\mathbb{C}}$ satisfying the size condition ${|K(x,y)|} \leq C{|x-y|}^{-d}$ and the regularity condition $${|K(x,y)-K(x',y)|}+{|K(y,x)-K(y,x')|} \leq \frac{C'}{{|x-y|}^d}\omega\left(\frac{{|x-x'|}}{{|x-y|}}\right),$$ whenever ${|x-x'|}\leq\frac{1}{2}{|x-y|},$ with the modulus of continuity $\omega\colon[0,1]\to{\mathbb{R}}_+$ that is continuous, increasing, subadditive, satisfies $\omega(0)=0$ and ${\big\|w\big\|_{\rm{Dini}}} :=
    \int_0^1\omega(t)\frac{dt}{t}<\infty.$

Given a $\gamma\in(0,1),$ we say that a collection of sets $\mathcal{F}$ is $\gamma$-sparse, if for all distinct elements $S,R\in\mathcal{F},$ there exist sets $E_S\subset S,E_R\subset R$ such that $E_S\cap E_R=\emptyset$ and ${|E_S|}>\gamma{|S|}.$

Let $T$ be as in definition \[CZO\]. We have the following maximal operators on $L^2({\mathbb{R}}^d):$

i)  the maximal operator $T_*f(x)=\sup_{\varepsilon>0}{\big|Tf1_{B(x,\varepsilon)^c}(x)\big|_{}},$

ii) the 