Dimension independent atomic decoposition for dyadic martingale $H^1$

TL;DR

This paper introduces a dimension-independent atomic decomposition for dyadic atomic $H^1$, providing sharp estimates for the associated maximal function, which advances understanding of harmonic analysis in high-dimensional settings.

Contribution

It presents a new atomic framework for dyadic $H^1$ that maintains equivalence with maximal function definitions regardless of dimension, with sharp norm estimates.

Findings

Dimension-independent atomic $H^1$ decomposition established.

Sharp $H^1 o L^1$ norm estimates for the maximal function provided.

Enhanced understanding of high-dimensional harmonic analysis achieved.

Abstract

We introduce atoms for dyadic atomic $H^1$ for which the equivalence between atomic and maximal function defnitions is dimension independent. We give the sharp, up to $\log(d)$ factor, estimates for the $H^1 \to L^1$ norm estimates for the special maximal function.