Local Hardy Spaces Associated with Inhomogeneous Higher Order Elliptic Operators

Let L be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this article, for all p ∈ (0, ∞) and L satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces hpL(R ) associated with L, which coincide with Goldberg’s local Hardy spaces h(R) for all p ∈ (0, ∞) when L ≡ −∆ (the Laplace operator). The authors also establish a real-variable theory of hpL(R ), 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 L ≡ − div(A∇) (the divergence form homogeneous second order elliptic operator). Moreover, the authors show that hpL(R ) coincides with the Hardy space H L+δ(R ) associated with the operator L + δ for all p ∈ (0, ∞), where δ is some positive constant depending on the ellipticity and off-diagonal estimates of L. As an application, the authors establish some mapping properties for the local Riesz transforms ∇(L+ δ)−1/2 on H L+δ(R ), where k ∈ {0, . . . , m} and p ∈ (0, 2].