A twosided linear estimate and a dyadic reduction of the UMD Conjecture

TL;DR

This paper introduces a dyadic shift operator and demonstrates its equivalence to the Hilbert transform's boundedness in Banach spaces, simplifying the UMD conjecture to dyadic operators.

Contribution

It defines a new dyadic shift operator and reduces the UMD conjecture to the analysis of simple dyadic operators.

Findings

Hilbert transform is $L^p$ bounded iff the dyadic shift is bounded

Dyadic shift has linear two-sided norm dependence

Reduces UMD conjecture to dyadic operators

Abstract

We define a time faithful dyadic shift operator of complexity one, that is an antisymmetric antiinvolution. We show that the Hilbert transform with values in a Banach space is $L^p$ bounded if and only if the dyadic shift is -- with a linear two sided norm dependence. The results reduce the famous UMD conjecture to a pair of simple dyadic operators.