# Regularity class of certain diffeomorphisms of the real line

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 $$\frac{\sqrt{\varphi'(x)\varphi'(y)}}{\varphi(x)-\varphi(y)}$$.