The dyadic Riesz vector II

TL;DR

This paper introduces a dyadic model operator for the Riesz vector, establishing linear bounds in L^p spaces and linking the model's boundedness to that of the actual Riesz vector in Banach space-valued functions.

Contribution

It constructs a dyadic Riesz vector model and proves its boundedness implies the boundedness of the classical Riesz vector, with a linear dependence constant of one.

Findings

Linear upper L^p bounds for the dyadic Riesz vector

Boundedness of the dyadic model implies boundedness of the Riesz vector

Results hold for Banach space-valued functions

Abstract

We derive a dyadic model operator for the Riesz vector. We show linear upper $L^p$ bounds for $1 < p < \infty$ between this model operator and the Riesz vector, when applied to functions with values in Banach spaces. By an upper bound we mean that the boundedness of the dyadic Riesz vector implies the boundedness of the Riesz vector. The same holds for single dyadic Riesz transforms and their continuous counterparts. The linear dependence is with constant one.