I can't prove the identity in Grafako's book Classical Fourier analysis page 256 directly: for $\xi$ fixed
$\operatorname{sgn}(\xi-y)\cdot\operatorname{sgn}(y)=1-\operatorname{sgn}(\xi)\cdot[\operatorname{sgn}(y)+\operatorname{sgn}(\xi-y)]$, where \operatorname{sgn}(x)=\begin{cases}1&\mbox{ if }x>0\\\ 0&\mbox{ if }x=0\\\ -1&\mbox{ if }x<0.\end{cases}
I would like to direct proof, without going through several cases.
Thank's