3 more Young functions to manage the now non-symmetric dual pairings from the terms $S_2,S_4.$ According to Definition \[sufdef\] the existing Young functions functions are replaced with ones satisfying $$\bar A \in B_{p'},\quad \bar B,\bar C\in B_p$$ and are supplemented with Young functions $D,E,F$ satisfying $$\bar D,\bar F\in B_p,\quad \bar E\in B_{p'}$$ and $S_{D,E}(b_1,b_2)+T_F(b_1,b_2) < \infty.$

\[altsuf\] Given a $q\in(2,\infty),$ adapting the proof of Theorem \[upperbound\] shows that if $b_1,b_2$ satisfy the conditions $S_{q+\varepsilon},T_{q+\varepsilon}$ for any $\varepsilon>0,$ then $C_bT:L^q\to L^q$ boundedly.

On the other hand, for $q\in(1,2),$ the conditions $S_p,T_p$ with $p\in(q,2)$ are not strong enough to conclude that $C_bT:L^q\to L^q$ boundedly. Indeed, if they were, then by duality and interpolation $C_bT:L^2\to L^2$ boundedly and Theorem \[lowerbound\] would imply the condition $S_2$. This gives a contradiction since by Proposition \[prop1\] there exists functions $\phi,\psi$ such that $S_p,T_p$ are satisfied and $S_2$ is not.

Conjecture and related examples
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In the last section we continue discussing the conditions $S_{A,B}, T_C,$ and their interdependence with the boundedness properties of the commutator on different $L^p$ spaces.

\[jnce-1\] There exists $b_1, b_2\in L^{\infty}_{\rm{loc}}({\mathbb{R}})$ such that $S_2(b_1, b_2)+T_2(b_1, b_2)<\infty$, but $S_{A,B}(b_1, b_2)=\infty$ and $T_C(b_1, b_2)=\infty$ for arbitrary Young functions $A,B,C$ with $\bar A, \bar B, 