Local Hardy spaces associated with inhomogeneous higher order elliptic operators

Abstract

Let [Formula: see text] be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this paper, for all [Formula: see text] and [Formula: see text] satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces [Formula: see text] associated with [Formula: see text], which coincide with Goldberg’s local Hardy spaces [Formula: see text] for all [Formula: see text] when [Formula: see text] (the Laplace operator). The authors also establish a real-variable theory of [Formula: see text], which includes their characterizations in terms of the local molecules, the square functions or the maximal functions, the complex interpolation and dual spaces. These real-variable characterizations on the local Hardy spaces are new even when [Formula: see text] (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that [Formula: see text] coincides with the Hardy space [Formula: see text] associated with the operator [Formula: see text] for all [Formula: see text], where [Formula: see text] is some positive constant depending on the ellipticity and the off-diagonal estimates of [Formula: see text]. As an application, the authors establish some mapping properties for the local Riesz transforms [Formula: see text] on [Formula: see text], where [Formula: see text] and [Formula: see text].