where the exceptional set is defined and control over the appearing terms is established. In addition, a comment is made about the rest of the proof, the details for which we refer the reader to the proof of Theorem 1.1 in [@LOR1] or [@L].

For an arbitrary integrable function $\psi\not=0$ on $Q$ define $$E_1(\psi) = \{ x\in Q\colon {|\psi(x)|} > \alpha \langle{|\psi|}\rangle_{3Q} \},\qquad
    E_2(\psi) = \{x\in Q\colon \mathcal{M}_{T,Q}\psi(x) > \alpha \langle{|\psi|}\rangle_{3Q} \}$$ and let the exceptional set be $$E = \bigcup_{i=1,2} E_i(f)\cup E_i(b_1f) \cup E_i(b_2f)\cup E_i(b_1b_2f).$$ Since the localized version of the grand maximal operator is controlled with the non-localized by $$\mathcal{M}_{T,Q}f \leq \mathcal{M}_Tf1_{3Q},$$ and by the well-known facts that $\mathcal{M},T_*\colon L^1\to L^{1,\infty}$ boundedly, it follows from the weak $(1,1)$ bounds implied by ii) of Lemma \[keylemma\] in conjunction with the local integrability of all functions in question that we may choose some $\alpha>0$ independent of the cube $Q$ so that ${|E|}\leq 2^{-(d+2)}{|Q|}$.

Taking a Calderón-Zygmund decomposition of the function $1_E$ at the height $2^{-(d+1)}$ yields a collection $\mathcal{F}$ of cubes satisfying: $$\sum_{P\in\mathcal{F}}{|P|} \leq\frac{1}{2}{|Q|}, \quad{|E\setminus\bigcup_{P\in\mathcal{F}}P|}=0\quad\mbox{ and }\quad P\cap E^c\not=\emptyset\qquad\forall P\in\mathcal{F}.$$ Then one decomposes $$\left( C_bTf1_{3Q}\right)1_Q =  \left( C_bTf1_{3Q}\right)1_{Q\setminus\cup P} + \sum_{P\in\mathcal{F}}\left( C_bTf1_{3Q\setminus 3P}\right)1_P + \sum_{P\in\mathcal{F}}\left( C_bTf1_{3P}\right)1_P$$ and uses the properties 