The dyadic and the continuous Hilbert transforms with values in Banach spaces

TL;DR

This paper establishes a linear relationship between the boundedness of the Hilbert transform and the dyadic Hilbert transform in Banach spaces, highlighting their interconnectedness in harmonic analysis.

Contribution

It proves that $L^p$ boundedness of the Hilbert transform in Banach spaces implies the boundedness of the dyadic Hilbert transform with a proportional norm.

Findings

Hilbert transform boundedness implies dyadic Hilbert transform boundedness

Linear relation between their operator norms

Advances understanding of harmonic analysis in Banach spaces

Abstract

We show that if the Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the dyadic Hilbert transform, with a linear relation of the norms.