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

We give an atomic decomposition of closed forms on Rn, 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 Hlog, appears naturally when considering products of functions in the Hardy space H1 and in BMO. As a main consequence of the atomic decomposition, we obtain a weak factorization of closed forms whose coefficients are in Hlog. Namely, a closed form in Hlog is the infinite sum of the wedge product between an exact form in the Hardy space H1 and an exact form in BMO. The converse result, which generalizes the classical div-curl lemma, is a consequence of [4]. As a corollary, we prove that the real-valued Hlog space can be weakly factorized.