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

TL;DR

This paper demonstrates that Petermichl's dyadic operator is a Calderón-Zygmund operator on a suitable space, establishing weighted boundedness results for its maximal truncation operator using dyadic A_p weights.

Contribution

It proves that Petermichl's operator fits the Calderón-Zygmund framework on a metric space of homogeneous type, extending weighted inequality theory to this operator.

Findings

Petermichl's operator is Calderón-Zygmund on a homogeneous space.

Weighted boundedness of the maximal truncation operator is established.

Dyadic A_p weights are effective for the operator's scale truncations.

Abstract

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 $\mathcal{P}^*$ of truncations of the singular integral. We show that dyadic $A_p$ weights are the good weights for the maximal operator $\mathcal{P}^*$ of the scale truncations of $\mathcal{P}$.