Let us summarize our main results in a theorem. For a function $f$ on the circle $S^1$, we let $H_{\bar{f}}$ denote the associated Hankel operator on the Hardy space $H^2(S^1)$ (see more below in Section \[hanekopdldl\] below). We let $\mathcal{M}^{(p)}_\psi$ denote the $p$-convexification of the Lorentz ideal $\mathcal{M}_\psi$ and $\mathcal{M}^{(p)}_{\psi,0}$ its separable subspace (see more in Subsection \[opidelalsos\] below), and let $\mathrm{Tr}_{\omega,\psi}: \mathcal{M}_\psi\to {{\mathbb C}}$ be the Dixmier trace associated with an exponentiation invariant extended limit $\omega$. We write $A\sim B$ if there is a universal constant $C>0$ such that $C^{-1}A\leq B\leq CB$. When saying universal, we still allow for a dependence on $p$ and $\psi$.

\[mainthmintro\] Let $p\geq 1$, $(\|\cdot\|_{B^{1/q}_{q,q},*})_{q\geq p}$ a family of norms on the Besov spaces $B^{1/q}_{q,q}(S^1)$ for $q\geq p$ satisfying the conditions of Corollary \[equivbesovgeneerl\], and $\psi:[0,\infty)\to [0,\infty)$ be an increasing concave function with regular variation of index $0$ satisfying $\psi(0)=0$, $\lim_{t\to \infty}\psi(t)=\infty$ and the conditions and . Then for any holomorphic function $f$ the following holds:

1.  $H_{\bar{f}}\in \mathcal{M}_{\psi}^{(p)}$ if and only if $\sup_{q>p}\frac{1}{\psi(\mathrm{e}^{(q-p)^{-1}})} \|f\|_{B^{1/q}_{q,q},*}^q<\infty$.

2.  For any exponentiation invariant extended limit $\omega$, $$\mathrm{Tr}_{\omega,\psi}(|H_{\bar{f}}|^p)\sim\lim_{q-p\to  \tilde{\omega}}\frac{1}{\psi(\mathrm{e}^{(q-p)^{-1}})} \|f\|_{B^{1/q}_{q,q},*}^q.$$ Here $\tilde{\omega}$ is defined as in Equation (see page ).

3.  It holds that $$\mathrm{d}_{\mathcal{M}_\psi}(|H_{\bar{f}}|^p,\mathcal{M}_{\psi,0}):=\inf_{A\in \mathcal{M}_{\psi,0}}\||H_{\bar{f}}|^p-A\|_{\mathcal{M}_\psi}\sim\limsup_{q\searrow p} \frac{1}{\psi(\mathrm{e}^{(q-p)^{-1}})}\|f\|_{B^{1/q}_{q,q},*}^q$$

