Use of abstract Hardy spaces, Real interpolation and Applications to bilinear operators

TL;DR

This paper develops an abstract framework for Hardy spaces using real interpolation, enabling advanced analysis of bilinear operators and singularities in a more general setting beyond Euclidean spaces.

Contribution

It introduces a refined real interpolation approach for Hardy spaces and applies bilinear interpolation theory to analyze bilinear operators abstractly.

Findings

Established interpolation results for Hardy spaces and Lebesgue spaces.

Applied the theory to study bilinear Calderon-Zygmund type operators.

Extended analysis of singular operators in an abstract framework.

Abstract

This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H^1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarify the use of Hardy spaces. Then with the help of the bilinear interpolation theory, we then give applications to study bilinear operators on Lebesgue spaces. These ideas permit us to study singular operators with singularities similar to those of bilinear Calderon-Zygmund operators in a far more abstract framework as in the euclidean case.