[0,\infty)$ be an increasing concave function with $\psi(0)=0$, $\lim_{t\to\infty} \psi(t)=\infty$. For later purposes of Dixmier trace computations, we often assume a condition which is slightly stronger than that in the original Dixmier paper. This condition is that the limit $$\label{apsicond}
A_{\psi}(\alpha):=\lim_{t\to \infty} \frac{\psi(t^\alpha)}{\psi(t)}\quad\mbox{exists for all $\alpha>1$}.$$ Since $\psi$ is increasing, $A_\psi(\alpha)\geq 1$ for all $\alpha$. Condition guarantees that $\psi$ has regular variation of index $0$. Recall that a function $\psi$ has regular variation of index $\rho\in {{\mathbb R}}$ if $$\lim_{t\to\infty}\frac{\psi(\lambda t)}{\psi(t)}=\lambda^\rho, \quad\forall \lambda>0.$$ By [@RegVar Theorem 1.8.2] we can without restrictions assume that $\psi$ is smooth. For the purpose of extrapolation results, the following condition on $\psi$ often comes into play: $$\label{Phi_as1}
\|\psi'\|_p \le C \psi(e^\frac1{p-1}), \ \forall p>1.$$

We define the Lorentz ideal $\mathcal{M}_\psi$ to consist of operators affiliated with $\mathcal{N}$ such that $$\|T\|_{\mathcal{M}_\psi}:=\sup_{t>0}\frac{1}{\psi(t)}\int_0^t\mu_T(s)\mathrm{d}s<\infty.$$ The norm $\|\cdot\|_{\mathcal{M}_\psi}$ makes $\mathcal{M}_\psi$ into a symmetrically normed operator ideal.

If the function $\psi$ satisfies condition , the ideal $\mathcal{M}_\psi$ carries a plethora of singular traces, with Dixmier traces being those of most relevance to this paper. For $\alpha\geq 1$ we define $P_\alpha:L^\infty(0,\infty)\to L^\infty(0,\infty)$ by $P_\alpha f(t):=f(t^\alpha)$. If $\omega\in L^\infty(0,\infty)^*$ is a state satisfying that $\omega(f)=0$ if $\lim_{t\to \infty}f(t)=0$ we say that $\omega$ is 