On the Calder\'{o}n-Zygmund structure of Petermichl's kernel. Weighted inequalities

We show that Petermichl’s dyadic operator $\mathcal{P}$ (S. Petermichl (2000), Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol) is a Calder'{o}n-Zygmund type operator on an adequate metric normal space of homogeneous type. As a consequence of a general result on spaces of homogeneous type, we get weighted boundedness of the maximal operator ${\mathcal{P}}^{\ast }$ of truncations of the singular integral. We show that dyadic $A_p$ weights are the good weights for the maximal operator ${\mathcal{P}}^{\ast }$ of the scale truncations of $\mathcal{P}$.