Independent sums of $H^1_n(\mathbb{T})$ and $H^1_n(\delta)$

TL;DR

This paper introduces a novel idempotent Fourier multiplier on the Hardy space on the bidisc, utilizing a new $L^1$ Stein martingale inequality and characterizes its range as an independent sum of dyadic $H^1_n$ spaces.

Contribution

It constructs a new Fourier multiplier on the Hardy space on the bidisc that cannot be derived from one-dimensional results, expanding the understanding of multipliers in several complex variables.

Findings

Developed a new $L^1$ Stein martingale inequality for special periodic filtrations.

Identified the range of the multiplier as an independent sum of dyadic $H^1_n$ spaces.

Showed the constructed operator's range is a complemented, invariant subspace of dyadic $H^1$.

Abstract

We construct a new idempotent Fourier multiplier on the Hardy space on the bidisc, which could not be obtained by applying known one dimentional results. The main tool is a new $L^1$ equivalent of the Stein martingale inequality which holds for a special filtration of periodic subsets of $\mathbb{T}$ with some restrictions on the functions involved. We also identify the isomorphic type of the range of the associated operator as the independent sum of dyadic $H^1_n$, which is known to be a complemented and invariant subspace of dyadic $H^1$.