T_{LP}f(z,n):=W_n*f(z),\quad\mbox{and}\\
T_{\mathbb{D}}:B^{1/q+}(S^1)&\to L^q(\mathbb{D},\mu), \quad T_{\mathbb{D}}f(z):=(1-|z|^2)^2f''(z).\end{aligned}$$ We define the spaces $\mathcal{M}_\psi^{(p)}(S^1\times {{\mathbb N}},\nu)\subseteq \cap_{q>p} L^q(S^1\times {{\mathbb N}},\nu)$ and $\mathcal{M}_\psi^{(p)}(\mathbb{D},\mu)\subseteq \cap_{q>p} L^q(\mathbb{D},\mu)$ from the families $(L^q(S^1\times {{\mathbb N}},\nu))_{q>p}$ and $(L^q(\mathbb{D},\mu)_{q>p}$, respectively, by means of extrapolation. For any exponentiation invariant extended limit $\omega$, we can define Dixmier traces $\mathrm{tr}_{\omega,\psi}:\mathcal{M}_\psi(S^1\times {{\mathbb N}},\nu)\to {{\mathbb C}}$ and $\mathrm{tr}_{\omega,\psi}:\mathcal{M}_\psi(\mathbb{D},\mu)\to {{\mathbb C}}$. We write $\mathrm{tr}_{\omega,\psi}$ to emphasize that the Dixmier trace is defined on a commutative von Neumann algebra. Applying Corollary \[equivalencescor\], we can reformulate Theorem \[Thm2\] as the statement that $${\rm Tr}_{\omega,\psi} |H_{\overline{f}}|^p\sim \mathrm{tr}_{\omega,\psi}(|T_{LP}f|^p)\sim\mathrm{tr}_{\omega,\psi}(|T_{\mathbb{D}}f|^p) .$$

The special case $p=2,4,6$ {#p246sec}
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A beautiful result of Janson-Upmeier-Wallstén [@JUW] computes the operator trace of $|H_{\overline{f}}|^p$ for $p=2,4,6$ in terms of a particular Besov norm. Indeed, [@JUW Theorem 1] states that for $p=2,4,6$ and $f$ holomorphic in $\mathbb{D}$ we have that $$\label{p246}
\mathrm{Tr}(|H_{\overline{f}}|^p)=c_p\int_{S^1\times S^1} \frac{|f(z)-f(w)|^{p}}{|z-w|^2}\mathrm{d}V(z,w),$$ where $c_2=1$, $c_4=\frac{1}{2}$ and $c_6=\frac{1}{6}$. In fact, [@JUW Theorem 1] states that $p=2,4,6$ are the only possible values for which an identity of this type could hold true.

\[SIdefn\] For $p>1$ we define $$\|f\|_{B^{1/p}_{p,p},SI}:=\left(\int_{S^1\times S^1} \frac{|f(z)-f(w)|^p}{|z-w|^2}\mathrm{d}V(z,w)\right)^{1/p}.$$

The reader should note that Equation is equivalent to the identity $$\|H_{\overline{f}}\|_{\mathcal{L}^p(L^2(S^1))}^p=c_p\|f\|_{B^{1/p}_{p,p},SI}^p, \quad\mbox{for $p=2,4,6$}.$$ The following norm equivalence is found in [@peller Appendix 2.6].

\[sivsdnorm\] For any $p_0\geq q_0>1$ 