New Abstract Hardy Spaces

TL;DR

This paper introduces an abstract framework for Hardy spaces using atomic decompositions, explores their properties, and applies the theory to maximal regularity and dual space characterization.

Contribution

It develops a new abstract construction of Hardy spaces that preserves key properties and extends their applications to maximal regularity and duality analysis.

Findings

Spaces maintain main Hardy space properties

Results on continuity into L^1 and interpolation

Partial characterization of dual spaces

Abstract

The aim of this paper is to propose an abstract construction of spaces which keep the main properties of the (already known) Hardy spaces H^1. We construct spaces through an atomic (or molecular) decomposition. We prove some results about continuity from these spaces into L^1 and some results about interpolation between these spaces and the Lebesgue spaces. We also obtain some results on weighted norm inequalities. Then we apply this abstract theory to the L^p maximal regularity. Finally we present partial results in order to understand a characterization of the duals of Hardy spaces.