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I care about the following class of homeomorphisms of $\mathbb R$, which I'll call $\mathcal C^?$.
For simplicity, let us restrict attention to compactly supported homeomorphisms (a homeomorphism $\varphi:\mathbb R\to \mathbb R$ is compactly supported if $\exists N>0$ such that $\forall x$ with $|x|>N$ we have $\varphi(x)=x$).

Let $\lambda$ be the Lebesgue measure on $\mathbb R$.

Definition
A compactly supported homeomorphism $\varphi:\mathbb R\to \mathbb R$ belongs to $\mathcal C^?$ if:
- ⓵ The measures $\lambda$ and $\varphi^\*(\lambda)$ are absolutely continuous with respect to each other.
- ⓶ The function $$ \frac{1}{x-y} \quad-\quad \frac{\sqrt{\varphi'(x)\varphi'(y)}}{\varphi(x)-\varphi(y)} $$ belongs to $L^2(\mathbb R^2)$. (Here, $\varphi'$ is a shorthand for the Radon–Nikodym derivative $d\varphi^\*(\lambda)/d\lambda$. It agrees with the derivative of $\varphi$ when the latter is smooth.)

I would like to have a more concrete description of $\mathcal C^?\subset \mathit{Homeo}(\mathbb R)$.

For example, I know of a $\mathcal C^1$ homeomorphism that is not $\mathcal C^?$.

Is it the case that $\mathcal C^? \subset \mathcal C^1$?


Motivation:
I'd like to understand the (projective) action of $Diff_c(\mathbb R)$ on the Fermionic Fock space $\bigoplus_{n\ge 0} \Lambda^nH$, and to which subgroup of $\mathit{Homeo}(\mathbb R)$ it extends. Here, $H\subset L^2\mathbb R$ is the Hardy space $H:=\mathcal F(L^2\mathbb R_{\ge 0})$ (and $\mathcal F$ is the Fourier transform). By the Segal quantization criterion, a unitary operator $\varphi$ on $L^2\mathbb R$ induces a corresponding operator on Fock space iff the difference between the Hilbert transform $\mathcal H$ and $\varphi \mathcal H\varphi^{-1}$ is Hilbert-Schmidt. In my case, the integral kernel of the Hilbert transform is $\frac{1}{x-y}$ (up to constant), and the integral kernel of $\varphi \mathcal H\varphi^{-1}$ is $\frac{\sqrt{\varphi'(x)\varphi'(y)}}{\varphi(x)-\varphi(y)}$.

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