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 paper, for all p∈(0,∞) and L satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces h_L^p(Rⁿ) associated with L, which coincide with Goldberg's local Hardy spaces h_L^p(Rⁿ) for all p∈(0,∞) when ≡-Δ (the Laplace operator). The authors also establish a real-variable theory of h_L^p(Rⁿ), 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 ≡-div(A∇) (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that h_L^p(Rⁿ) coincides with the Hardy space H^p_L(Rⁿ) associated with the operator L + δ for all p∈(0,∞), where δ is some positive constant depending on the ellipticity and the off-diagonal estimates of L. As an application, the authors establish some mapping properties for the local Riesz transforms ∇^k(L+δ))^-1/2 on H_L+δ^p(Rⁿ), where k {0,...,m} and p∈(0, 2].