A simple sharp weighted estimate of the dyadic shifts on metric spaces with geometric doubling

TL;DR

This paper provides a simple polynomial estimate for the norm of weighted dyadic shifts on metric spaces with geometric doubling, which is linear in the weight norm, advancing the understanding of Calderón-Zygmund operators.

Contribution

It introduces a straightforward polynomial estimate for dyadic shifts on metric spaces with geometric doubling, enabling linear bounds for Calderón-Zygmund operators.

Findings

Polynomial estimate of dyadic shift norm is linear in weight norm.

Supports decomposition of Calderón-Zygmund operators into dyadic shifts.

Facilitates linear weighted bounds for Calderón-Zygmund operators.

Abstract

We give a short and simple polynomial estimate of the norm of weighted dyadic shift on metric space with geometric doubling, which is linear in the norm of the weight. Combined with the existence of special probability space of dyadic lattices built in A. Reznikov, A. Volberg, "Random "dyadic" lattice in geometrically doubling metric space and $A_2$ conjecture", arXiv:1103.5246, and with decomposition of Calder\'on-Zygmund operators to dyadic shifts from Hyt\"onen's "The sharp weighted bound for general Calder\'on-Zygmund operators", arXiv:1007.4330 (and later T. Hyt\"onen, C. P\'erez, S. Treil, A. Volberg, "A sharp estimated of weighted dyadic shifts that gives the proof of $A_2$ conjecture", arXiv 1010.0755), we will be able to have a linear (in the norm of weight) estimate of an arbitrary Calder\'on-Zygmund operator on a metric space with geometric doubling. This will be published separately.