and there is a constant $C>0$ such that $$C^{-1}\|f\|_{B^{1/p}_{p,p}}\leq \|f\|_{B^{1/p}_{p,p},SI}\leq C\|f\|_{B^{1/p}_{p,p}}, \quad \forall p\in [q_0,p_0].$$

The result of Janson-Upmeier-Wallstén together with Theorem \[pellerhem\] and Proposition \[sivsdnorm\] allow us to deduce the following proposition.

\[equiforjuw\] There are constants $0<r< R<\infty$ and measurable functions $c_0,c_1:[3/2,8]\to [r,R]$ such that $$c_0(p)\|f\|_{B^{1/p}_{p,p},SI}\leq \|H_{\overline{f}}\|_{\mathcal{L}^p(L^2(S^1))}\leq c_1(p)\|f\|_{B^{1/p}_{p,p},SI}.$$ Moreover, we can choose $c_0$ and $c_1$ so that $$\lim_{h\to 0} c_0(p+h)^{\frac{1}{p}}=\lim_{h\to 0} c_1(p+h)^{\frac{1}{p}}=c_p\quad\mbox{for $p=2,4,6$}.$$

For $p>1$ and the scale of spaces $(B^{1/q}_{q,q}(S^1))_{q\in [p,p+1]}$ we let $B_{p,\psi}(S^1)$ denote the associated extrapolation space. Using Corollary \[equivalencescor\], we deduce the following theorem from Proposition \[equiforjuw\].

For $p=2,4,6$, and a holomorphic $f\in B_{p,\psi}(S^1)$, we have that $$\mathrm{Tr}_{\omega,\psi}(|H_f|^p)= c_p\lim_{h\to  \tilde{\omega}} \frac{1}{\psi(\mathrm{e}^{1/h})}\int_{S^1\times S^1} \frac{|f(z)-f(w)|^{p+h}}{|z-w|^2}\mathrm{d}V(z,w),$$ where $c_2=1$, $c_4=\frac{1}{2}$ and $c_6=\frac{1}{6}$.

The special case $p=2$ and $f\in C^{1/2}(S^1)$ was considered in [@GG], where explicit formulas for $\mathrm{Tr}_\omega(|H_{\overline{f}}|^2)$ was given in terms of the Fourier series of $f$.

Non-measurability {#Non-meas}
=================

The estimates for Dixmier traces will allow us to construct an abundance of non-measurable Hankel operators by means of lacunary Fourier series. Our approach is based on results from [@EZ Section 6]. For $p\in [1,\infty)$ and $c\in \ell^\infty({{\mathbb N}})$ we define the function $\gamma_{p,c}$ on $S^1$ by $$\gamma_{p,c}(z):=\sum_{j=0}^\infty 2^{-j/p}c_j z^{2^j}.$$ The function $\gamma_{p,c}$ is holomorphic in $\mathbb{D}$. We 