Atomic decomposition and weak factorization in generalized Hardy spaces of closed forms

TL;DR

This paper develops an atomic decomposition for closed forms in generalized Hardy spaces of Musielak-Orlicz type, leading to a weak factorization result that generalizes classical div-curl lemmas.

Contribution

It introduces a novel atomic decomposition for closed forms in Musielak-Orlicz Hardy spaces and establishes a weak factorization theorem extending classical results.

Findings

Atomic decomposition for closed forms in H log spaces.

Weak factorization of closed forms as wedge products of Hardy and BMO forms.

Generalization of the div-curl lemma to Musielak-Orlicz Hardy spaces.

Abstract

We give an atomic decomposition of closed forms on R n , the coefficients of which belong to some Hardy space of Musielak-Orlicz type. These spaces are natural generalizations of weighted Hardy-Orlicz spaces, when the Orlicz function depends on the space variable. One of them, called H log , appears naturally when considering products of functions in the Hardy space H 1 and in BM O. As a main consequence of the atomic decomposition, we obtain a weak factorization of closed forms whose coefficients are in H log. Namely, a closed form in H log is the infinite sum of the wedge product between an exact form in the Hardy space H 1 and an exact form in BM O. The converse result, which generalizes the classical div-curl lemma, is a consequence of [4]. As a corollary, we prove that the real-valued H log space can be weakly factorized.