A Decomposition for Hardy Martingales. Part II

TL;DR

This paper establishes inequalities for dyadic perturbations of Hardy martingales, providing new estimates for their $L^1$ distance and revisiting Bourgains' embedding of $L^1$ into a quotient space.

Contribution

It introduces Davis and Garsia inequalities for perturbed Hardy martingales and applies them to analyze $L^1$ distances, offering new insights into martingale embeddings.

Findings

Proved Davis and Garsia inequalities for dyadic perturbations.

Estimated $L^1$ distance of dyadic martingales to Hardy martingales.

Revisited Bourgains embedding of $L^1$ into $L^1 / H^1$.

Abstract

We prove Davis and Garsia Inequalities for dyadic perturbations of Hardy Martingales. We apply those to estimate the $L^1 $ distance of a dyadic martingale to the class of Hardy martingales. We revisit Bourgains embedding of $L^1$ into the quotient space $ L^1 / H^1 . $