If $f\in L^{\infty}(\mathbb{R})$, do we have the following formula?
$P_{y}*(Hf)(x)=H(P_{y}*f)(x)$?
What I know is that Hilbert transform is a bounded operator from $L^{\infty}$ to $\operatorname{BMO}$ (also $\operatorname{BMO}$ to $\operatorname{BMO}$), and Poisson integral of a $\operatorname{BMO}$ for any fixed $y$ is also a $\operatorname{BMO}$ function. But I have no idea how to interchange the order of a principal value integral, any help will be appreciated.