&\leq 2\|\xi_j\|_2 \|P\|.
\end{aligned}$$

Note that Lemma \[Dab-estimates\] is actually a slight extension of Lemma 12 in [@Dab-Gamma], since we added in and each time the second estimates. In fact, they can be easily deduced from the first estimates by using the more general identity $$\label{duality}
(\tau\otimes{\operatorname{id}})(P_1 (\partial_i P_2))^\ast = ({\operatorname{id}}\otimes\tau)((\partial_i P_2^\ast) P_1^\ast)$$ for all $P_1,P_2\in{\mathbb{C}}\langle X_1,\dots,X_n\rangle$, which itself can easily be checked on monomials.

Moreover, we note that thanks to , the inequalities in can be rewritten in the following way: $$\label{conjugate_norms}
\begin{aligned}
\|\partial_j^\ast(P \otimes 1)\|_2 &\leq \|\xi_j\|_2 \|P\|\\
\|\partial_j^\ast(1 \otimes P)\|_2 &\leq \|\xi_j\|_2 \|P\|
\end{aligned}$$

We will need the following extension of and of .

\[key-estimates\] In the situation described above, the following holds true:

-   For all $Y_1,Y_2\in {\mathbb{C}}\langle X_1,\dots,X_n\rangle$, we have for $i=1,\dots,n$ that $$\|\partial_i^\ast(Y_1 \otimes Y_2)\|_2 \leq 3 \|\xi_i\|_2 \|Y_1\| \|Y_2\|.$$

-   For all $Y_1,Y_2\in {\mathbb{C}}\langle X_1,\dots,X_n\rangle$, we have for $i=1,\dots,n$ that $$\begin{aligned}
    \|({\operatorname{id}}\otimes\tau)((\partial_i Y_1) 1 \otimes Y_2)\|_2 & \leq 4 \|\xi_i\|_2 \|Y_1\| \|Y_2\|,\\
    \|(\tau\otimes{\operatorname{id}})(Y_1 \otimes 1 (\partial_i Y_2))\|_2 & \leq 4 \|\xi_i\|_2 \|Y_1\| \|Y_2\|.\end{aligned}$$

According to Voiculescu’s formula, which we recalled in Lemma \[Voi-formula\], we have for all $Y_1,Y_2 \in {\mathbb{C}}\langle X_1,\dots,X_n\rangle$ that $$\begin{aligned}
 \partial_i^\ast (Y_1 \otimes Y_2)
 &= Y_1 \xi_i Y_2 - m_1({\operatorname{id}}\otimes\tau\otimes{\operatorname{id}})(\partial_i 