First, $$C_1f(x)=p.v.\int_\mathbb{R}\frac{A(x)-A(y)}{(x-y)^2}f(y)dy$$ Here, $C_1$ is the Calderon's first commutator. The author says, the $L^p$ boundedness of $C_1$ is coincides with $[|D|,A]$ where $$[|D|,A]f=|D|(Af)-A(|D|f)$$, and $\widehat{|D|f}=2\pi|\xi|\hat{f}(\xi)$. The following is what I have tried.
First I assumed that $$|D|Af-A|D|f=(|D|A)f$$ holds. So $$\begin{split}\widehat{(|D|A)f}(\xi)&=(|D|A*f)(\xi)\\&=\int_\mathbb{R}|\eta|\hat{A}(\eta)\hat{f}(\xi-\eta)d\eta.\end{split}$$
How can I improve this to have it coincides with $C_1$?
A is a function such that $A'\in L^\infty(\mathbb{R})$
From Classical and Multilinear Harmonic Analysis - Schlag, Muscalu