Div–curl type estimates for elliptic systems of complex vector fields

In this work, we extend the classical div–curl lemma proved by Coifman, Lions, Meyer and Semmes to the setup of elliptic complexes of complex vector fields with smooth coefficients. The tools include local versions of the Gagliardo–Nirenberg estimate and Rellich–Kondrachov compactness theorem for compactly supported distributions in the local Hardy space hp(RN)hp(RN) that we prove to hold for 0