important to keep track of the norms used on the Besov spaces. Let us state a general result regarding the estimates of Dixmier traces of Hankel operators. This statement is a direct consequence of Corollary \[equivalencescor\] and Theorem \[pellerhem\].

\[equivbesovgeneerl\] Let $p\geq 1$, and $\psi:$ be a function as in Corollary \[equivalencescor\]. Assume that $(\|\cdot\|_{B^{1/q}_{q,q},*})_{q\geq p}$ is a family of norms on the Besov spaces $B^{1/q}_{q,q}(S^1)$ for $q\geq p$ such that there is a $p_0>p$ and a constant $C_0>0$ such that $$C_0^{-1} \|f\|_{B^{1/q}_{q,q},*}\leq \|f\|_{B^{1/q}_{q,q}}\leq C_0 \|f\|_{B^{1/q}_{q,q},*}, \quad \forall q\in [p,p_0].$$ Then for any holomorphic function $f$, $$H_{\bar{f}}\in \mathcal{M}_\psi^{(p)}\quad\Leftrightarrow \sup_{q>p}\frac{\|f\|_{B^{1/q}_{q,q},*}^q}{\psi(\mathrm{e}^{(q-p)^{-1}})} <\infty.$$ Moreover, there is a constant $C>0$ (independent of $f$) such that for any exponentiation invariant extended limit $\omega$, $$C^{-1}\lim_{q-p\to  \tilde{\omega}}\frac{1}{\psi(\mathrm{e}^{(q-p)^{-1}})} \|f\|_{B^{1/q}_{q,q},*}^q\leq \mathrm{Tr}_{\omega,\psi}(|H_{\bar{f}}|^p)\leq C\lim_{q-p\to  \tilde{\omega}}\frac{1}{\psi(\mathrm{e}^{(q-p)^{-1}})} \|f\|_{B^{1/q}_{q,q},*}^q.$$ Finally, for any holomorphic function $f\in \cap_{q>p} B^{1/q}_{q,q}(S^1)$ $$\mathrm{d}_{\mathcal{M}_\psi}(|H_{\bar{f}}|^p,\mathcal{M}_{\psi,0})\sim\limsup_{q\searrow p} \frac{1}{\psi(\mathrm{e}^{(q-p)^{-1}})}\|f\|_{B^{1/q}_{q,q},*}^{q}$$

Corollary \[equivbesovgeneerl\] can be applied to a variety of different norms on the scale of Besov spaces. Let $f$ be a function on $S^1$ extending holomorphically to the unit disc and $p\in [1,\infty)$. By an abuse of notation, we identify $f$ with its holomorphic extension $f:\mathbb{D}\to {{\mathbb C}}$, where $\mathbb{D}$ denotes the unit disc. Let $\mu$ denote the measure on $\mathbb{D}$ given by $\mathrm{d}\mu(z)=(1-|z|^2)^{-2}\mathrm{d}m(z)$ 