C B$ for some constant $C>0$ depending only on the dimension of the underlying space, on integration exponents and on other concurrently unimportant absolute constants appearing in the assumptions. Then naturally $A \sim B$, if $A \lesssim B$ and $B \lesssim  A.$ Subscripts on constants ($C_{a,b,c,...}$) signify their dependence on those subscripts.

We also denote the space $L^p({\mathbb{R}}^d)$ with $L^p.$

Integral average is by dash or brackets: $\frac{1}{{|Q|}}\int_Qf = {\dashint_{Q} {f}} = \langle f\rangle_Q.$

Acknowledgements {#acknowledgements .unnumbered}
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We thank Henri Martikainen for posing research questions for this paper and for valuable discussions and comments.

The conditions $S_p,T_p$ and $S_{A,B},T_C$
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Before proceeding any further, let us precisely define the commutator $[b_2,[b_1,T]].$

\[comdef\] Let $b_i,$ $i=1,2,$ be such that $b_1,b_2,b_1b_2 \in L^2_{loc}({\mathbb{R}}^d)$ and denote $b=(b_1,b_2).$ With $T$ being an operator on $L^2({\mathbb{R}}^d)$, the commutator $C_bT$ on $L^{\infty}_c({\mathbb{R}}^d)$ is defined as $$\label{basicdef}
    C_bT= [b_2,[b_1,T]]$$ where $[A,B] = AB-BA$ for any two operations $A,B,$ and $b_if(\cdot) = b_i(\cdot)f(\cdot).$

We deal with the second order commutator $[b_2,[b_1,T]]$ but our results concerning sufficient conditions could just as well be formulated in the higher order cases.

It follows by the John-Nirenberg inequality that if $b_1,b_2\in{\operatorname{BMO}},$ then the conditions $S_p,T_q$ hold for all $p,q\geq1.$ Hence, a 