can compute that $$\Phi(t)=2^{-j/p} c_j, \quad t\in [2^j-1,2^{j+1}-1).$$ Therefore, $\|\gamma_{p,c}\|_{B^{1/p}_{p,p}}\sim \|\Phi\|_{L^p(0,\infty)}\sim \|c\|_{\ell^p({{\mathbb N}})}$. Moreover, we can as in [@EZ Page 351] compute that for $t\in [2^j-1,2^{j+1}-1)$ $$\label{equistuff}
\frac{\sum_{k=0}^{j-1} |c_k|^p}{\psi(2^j-1)}\lesssim \frac{1}{\psi(t)}\int_0^t \Phi(t)^p\mathrm{d}t\lesssim\frac{\sum_{k=0}^{j} |c_k|^p}{\psi(2^{j+1}-1)}.$$

Define the function $\tilde{\psi}(t):=\psi(2^t-1)$. This is again an increasing function with $\tilde{\psi}(0)=0$ and $\lim_{t\to \infty}\tilde{\psi}(t)=\infty$. If $\psi$ satisfies , then $\tilde{\psi}$ has regular variation of index $k_\psi:=\log A_\psi(e)$. We write $\mathfrak{m}_{\tilde{\psi}}^{(p)}({{\mathbb N}}):=\mathcal{M}_{\tilde{\psi}}^{(p)}(\ell^\infty({{\mathbb N}}))$. The inequalities imply that $$\|c\|_{\mathfrak{m}_{\tilde{\psi}}^{(p)}}^p\sim \sup_{t>0}\frac{1}{\psi(t)}\int_0^t \Phi(t)^p\mathrm{d}t\sim \|\gamma_{p,c}\|_{B_{p,\psi}}^p,$$ so the map $c\mapsto \gamma_{p,c}$ defines a continuous mapping $$\gamma:\mathfrak{m}_{\tilde{\psi}}^{(p)}({{\mathbb N}})\to B_{p,\psi}.$$

It follows from Theorem \[Thm2\] and the inequalities that for any exponentiation invariant extended limit $\omega$ we have $$\begin{aligned}
\mathrm{Tr}_{\omega,\psi}(|H_{\overline{\gamma_{p,c}}}|^p) &\sim \lim_{t\to \omega} \frac{1}{\psi(t)}\int_0^t \Phi(t)^p\mathrm{d}t \sim \lim_{t\to \omega} \frac{\sum_{k=0}^{\lfloor \log_2 t \rfloor} |c_k|^p}{\psi(2^{\lfloor \log_2 t \rfloor+1}-1)} \\
&= \lim_{t\to \omega\circ \log_2} \frac{\sum_{k=0}^{\lfloor  t \rfloor} |c_k|^p}{\psi(2^{\lfloor t \rfloor+1}-1)}=\lim_{t\to \omega\circ \log_2} \frac{\sum_{k=0}^{\lfloor  t \rfloor} |c_k|^p}{\tilde\psi( t )} .\end{aligned}$$ Denote $$\begin{aligned}
\label{small_DT}
\mathrm{tr}_{\omega\circ \log_2,\tilde{\psi}}(x):= \lim_{t\to\omega\circ \log_2} \frac{\sum_{k=0}^{\lfloor  t \rfloor} x_k}{\tilde\psi( t )}, \quad x \in \mathfrak{m}_{\tilde{\psi}}({{\mathbb N}})_+.\end{aligned}$$ Here we note that $\mathrm{tr}_{\omega\circ \log_2,\tilde{\psi}}$ extends to a singular linear functional on $\mathfrak{m}_{\tilde{\psi}}({{\mathbb N}})$ because it is the Dixmier trace $\mathrm{tr}_{\omega,\psi}$ on $\mathfrak{m}_{\psi}({{\mathbb N}})$ pulled back along the isometric order preserving embedding $\mathfrak{m}_{\tilde{\psi}}({{\mathbb N}})\hookrightarrow \mathfrak{m}_{\psi}({{\mathbb N}})$ defined by $b=(b_n)_{n\in {{\mathbb 