Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators

Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coeffcients and (p_-(L), p₊(L)) be the maximal interval of exponents q ϵ [1, ∞] such that the semigroup {e^-tL}t>0 is bounded on L^q(Rⁿ). In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces H^pL(Rⁿ) for all p ϵ (0, p₊ (L)), which when p = 1, answers a question asked by Deng, Ding and Yao in [21]. Moreover, the authors characterize H_L^p (Rⁿ) via various versions of square functions and Lusin-area functions associated to the operator L.