\rangle}_{ C, I_k}\\
    &=\lim_{k\rightarrow \infty} (1-k^{-1})  {\big \langle}|b_2- \langle b_2\rangle_{I_k}| 1_{(0,1)}{\big \rangle}_{ C, I_k}\\
&\simeq \lim_{k\rightarrow \infty} (1-k^{-1})k^{\frac 12}\log(e+k)^{-\frac 12}\log\log(e^e+k)^{-\frac 34} k^{-\frac 12}\log(e+k)^{\frac{1+\alpha}2}=\infty.
    \end{aligned}$$

\[notsuff\] The conditions $S_{2+\varepsilon},T_{2+\varepsilon}$ are not precise enough to yield a characterization of $C_bH:L^2\to L^2.$

The iterated commutator $C_bH$ of Theorem \[lastexample\] is bounded on $L^2.$ We show that the conditions $S_{2+\varepsilon}(b_1,b_2)$ and $T_{2+\varepsilon}(b_1,b_2)$ are not satisfied for any $\varepsilon>0$. To see this, it is enough to notice that for all log -bumps $A,B,C$ with $\bar A,\bar B,\bar C \in B_2,$ we have $t^{2+\varepsilon} \succeq A(t),B(t),C(t)$ by which by Proposition \[prop:young\] iv) and the estimates in Theorem \[lastexample\] it follows that $$\langle{|b_1-\langle b_1\rangle_{J_k}|}^{2+\varepsilon}\rangle_{J_k}^{\frac{1}{2+\varepsilon}}    \langle{|b_2-\langle b_2\rangle_{J_k}|}^{2+\varepsilon}\rangle_{J_k}^{\frac{1}{2+\varepsilon}} \gtrsim \langle{|b_1-\langle b_1\rangle_{J_k}|}\rangle_{A,J_k} \langle{|b_2-\langle b_2\rangle_{J_k}|}\rangle_{B,J_k} \to \infty,$$ and $$\langle{|b_1-\langle b_1\rangle_{I_k}|}^{2+\varepsilon}{|b_2-\langle b_2\rangle_{I_k}|}^{2+\varepsilon}\rangle_{I_k}^{\frac{1}{2+\varepsilon}} \gtrsim  \langle{|b_1-\langle b_1\rangle_{I_k}|}{|b_2-\langle b_2\rangle_{I_k}|}\rangle_{C,I_k} \to \infty,$$ as $k\rightarrow\infty$, showing that $S_{2+\varepsilon}(b_1,b_2) = \infty$ and $T_{2+\varepsilon}(b_1,b_2)=\infty.$

\[altnotsuff\] The commutator of Theorem \[lastexample\] is bounded on $L^2$ and unbounded on all $L^p,$ $p\in(1,\infty)\setminus\{2\}.$

Let $p>2,$ $q\in(2,p)$ and $f(x) = x^{-1/q}1_{[100,\infty)}(x)\in L^p.$ For all $x\in[-1,1],$ $$\begin{split}
    &|C_bHf(x)| = \left|\int(b_1(x)-b_1(y))(b_2(x)-b_2(y))\frac{f(y)}{x-y}{\,\mathrm{d}}y\right| \\
    &\sim \int_{100}^\infty \frac{y^{\frac 12}}{\log(e+y)^{\frac 12}\log\log(e^e+y)^{\frac 34}}\frac{y^{-\frac{1}{q}}}{y}{\,\mathrm{d}}y = \infty,
    \end{split}$$ showing that $C_bH:L^p\not\to L^p.$ It follows by duality that also $C_bH: L^{p'}\not\to L^{p'}.$

Alternatively, we can prove Corollary \[notsuff\] by Corollary \[altnotsuff\]. Indeed, if 