of the collection $\mathcal{F},$ Lemma \[keylemma\] and that the commutator is unchanged modulo constants in the functions $b_1,b_2$ to derive $$\begin{split}
        {\Big|C_bTf1_{3Q}\Big|_{}} 1_Q&\leq C_{T,d}\Big( {|b_2-\langle b_2\rangle_{3Q}|}{|b_1-\langle b_1\rangle_{3Q}|}\langle{|f|}\rangle_{3Q} \\
        &+ {|b_2-\langle b_2\rangle_{3Q}|}\langle{|b_1-\langle b_1\rangle_{3Q}|}{|f|}\rangle_{3Q} \\
        &+ {|b_1-\langle b_1\rangle_{3Q}|}\langle{|b_2-\langle b_2\rangle_{3Q}|}{|f|}\rangle_{3Q} \\
        &+ \langle{|b_1-\langle b_1\rangle_{3Q}|}{|b_2-\langle b_2\rangle_{3Q}|}{|f|}\rangle_{3Q} \Big)1_Q + \sum_{P\in\mathcal{F}}{\Big|C_bTf1_{3P}\Big|_{}}1_P .
            \end{split}$$ From this situation one first iterates the above estimate with the last term and then transfers the limit construction from the local to the global.

\[upperbound\] Assume that a pair of functions $b_1\in L^A_{\rm{loc}}({\mathbb{R}}^d)$ and $b_2\in L^B_{\rm{loc}}({\mathbb{R}}^d)$ with $b_1^2, b_2^2\in  L^{C}_{\rm{loc}}({\mathbb{R}}^d)$ satisfy the conditions $T_C$ and $S_{A,B}$ for some Young functions $A,B,C$ with $\bar A, \bar B, \bar C\in B_2$, then $$\begin{split}
    S_i\colon L^2({\mathbb{R}}^d)\cap L^3_c({\mathbb{R}}^d) \longrightarrow L^2({\mathbb{R}}^d), \qquad i=1,2,3,4
    \end{split}$$ boundedly.

Especially, it follows with a standard density argument by Theorem \[sparsedom\] that $$\begin{split}
    C_bT\colon L^2({\mathbb{R}}^d) \longrightarrow L^2({\mathbb{R}}^d)
    \end{split}$$ boundedly when notation and assumptions are retained.

The pairs of terms $S_1,S_4$ and $S_2,S_3$ are symmetric with respect to dual pairings. Hence, we show the estimate in the two distinct cases of $S_1$ and $S_3.$ By duality it is enough to estimate the pairings $\langle S_i(f),\psi\rangle$.

First, for the term $S_1$ we only use the assumptions involving the functions $A,B.$ By sparseness we get 