Higher order div-curl type estimates for elliptic linear differential operators on localizable Hardy spaces

Abstract

In this PhD thesis, we investigate div-curl type estimates associated with linear partial differential operators on Hardy-Sobolev spaces. The main result establishes a local extension of div-curl estimates for elliptic homogeneous linear partial differential operators with smooth complex coefficients in localizable Hardy-Sobolev spaces, inspired by the classical result due to Coifman, Lions, Meyer and Semmes. This version recovers known results in the literature concerning first-order operators associated with elliptic systems of complex vector fields. As key tools, we develop a new Calderón-Zygmund decomposition and a Poincaré-type inequality in localizable Hardy-Sobolev spaces. A second main result presents a global nonhomogeneous version of div-curl type estimates in the localizable Hardy space for p = 1, associated with complex vector fields with constant coefficients. As an application, we provide a new characterization of the local bmo space in Euclidean space associated with div-curl terms.