Young functions $A, B, C$ such that $\bar B,\bar C\in B_p,$ $ \bar A\in B_{p'}$ and a pair of complex valued functions $b_1\in L^A_{\rm{loc}}({\mathbb{R}}^d), b_2\in L^B_{\rm{loc}}({\mathbb{R}}^d)$, we say that the joint condition $S_{A,B}$ holds if $$\begin{aligned}
        S_{A,B}(b_1, b_2):= \sup_Q {\big \langle}|b_1- \langle b_1\rangle_Q|{\big \rangle}_{ A, Q}{\big \langle}|b_2- \langle b_2\rangle_Q|{\big \rangle}_{ B, Q}<\infty,
        \end{aligned}$$ and for $b_1^2, b_2^2\in L^{C}_{\rm{loc}}({\mathbb{R}}^d)$, we say that the joint condition $T_{C}$ holds if $$\label{ms}
        T_{C}(b_1, b_2):=  \sup_Q {\big \langle}|b_1- \langle b_1\rangle_Q| |b_2- \langle b_2\rangle_Q| {\big \rangle}_{ C, Q}<\infty.$$

We remark immediately, that in Theorem \[jnce-1\] we find a commutator that is unbounded on $L^2$ and that satisfies the conditions $S_2+T_2$ but fails the conditions $S_{A,B} + T_C$ for all Young functions $\bar A,\bar B,\bar C\in B_2.$

Necessary conditions
====================

We move on to derive the lower bounds $T_2$ and $S_2$ for the iterated commutator taken with the Riesz transforms. Our proof separates into two cases, to odd and even dimensions. Later we see that the conditions $T_2$ and $S_2$ are not strong enough to imply the $L^2$ boundedness of the commutator, however.

\[rieszlemma\] Let $R_j$ be the j:th Riesz transform on ${\mathbb{R}}^d,$ $j=1,\dots,d,$ $f_1,f_2\in L^{\infty}_c$ and $b_1,b_2,b_1b_2 \in L^1_{loc}.$ Under these assumptions, for all cubes $Q,$ 