Multiplication between elements in Martingale Hardy spaces and their duals

TL;DR

This paper develops bilinear decompositions for products of elements in martingale Hardy spaces and their duals, utilizing martingale paraproducts, with extensions to spaces of homogeneous type.

Contribution

It introduces continuous bilinear decompositions for martingale Hardy spaces and their duals, based on martingale paraproducts, including results for spaces of homogeneous type.

Findings

Established bilinear decompositions for $H^p$ and dual spaces.

Extended results to dyadic martingales on spaces with doubling measures.

Provided new tools for analyzing products in martingale Hardy spaces.

Abstract

In this paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces $ H^p\ (0<p\leqslant 1) $ and functions in their dual spaces. Our decompositions are based on martingale paraproducts. As a consequence of our work, we also obtain analogous results for dyadic martingales on spaces of homogeneous type equipped with a doubling measure.