$\|T(x)\|_{\mathcal{L}^{p+h}}$.* We will utilize this fact below for Hankel operators.

In part A of Corollary \[equivalencescor\], we can obtain equivalences that are independent of $\omega$. Indeed the upper and lower bounds on $c_0$ and $c_1$ implies that under the assumptions of of Corollary \[equivalencescor\], $$r\lim_{h\to  \tilde{\omega}} \frac{1}{\psi(\mathrm{e}^{1/h})}\|x\|_{X_{h}}^{p+h}\leq \mathrm{Tr}_{\omega,\psi}(|T(x)|^p)\leq R\lim_{h\to  \tilde{\omega}} \frac{1}{\psi(\mathrm{e}^{1/h})}\|x\|_{X_{h}}^{p+h}$$

Hankel operators and Peller’s characterization {#hanekopdldl}
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We now turn our focus to Hankel operators on the Hardy space. The reader can recall that the Hardy space $H^2(S^1)\subseteq L^2(S^1)$ is defined as the subspace of functions with a holomorphic extension to the interior of the unit disc. We here consider $S^1$ to be the boundary of the unit disc in the complex plane. The orthogonal projection $P:L^2(S^1)\to H^2(S^1)$ is called the Szegö projection. For $f\in L^\infty(S^1)$, the associated Hankel operator is defined as $$H_f:=(1-P)fP.$$ Clearly, if $f$ is the restriction of holomorphic function in the unit disc, $H_f=0$. In fact, $H_f$ is a well defined bounded operator for $f\in BMO(S^1)$. The space of symbols $f$ for which $H_f\in \mathcal{L}^p(L^2(S^1))$ has been characterized in terms of Besov spaces by Peller [@peller]. We let $B^{1/p}_{p,p}(S^1)$ denote the Besov space on $S^1$, we will review this space and various equivalent norms on 