Multiplication between elements in martingale Hardy spaces and their dual spaces

In the first part of the paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces H^p(0<p⩽1) and functions in their dual spaces. Our decompositions are based on martingale paraproducts. The second part of the paper concerns applications of the method developed for the classical martingales in the first part. In particular, we build a connection between Hardy spaces on spaces of homogenous type equipped with a doubling measure and Hardy spaces with respect to the corresponding dyadic martingales. Using the method introduced in the first part, we obtain analogous results for dyadic martingales on spaces of homogenous type, which, thanks to the aforementioned connection, yield conclusions for products between elements in Hardy spaces and those in their duals on spaces of homogenous type. The key property of martingale Hardy spaces in the study is that they admit the atomic decomposition, for which we provide an interpretation via duality.