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 coefficients and (p-(L), p(+) (L)) be the maximal interval of exponents q epsilon [1, infinity] such that the semigroup {e(-tL)}(t>0) is bounded on L-q(R-n). In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces H-L(p)(R-n) for all p epsilon (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) (R-n) via various versions of square functions and Lusin-area functions associated to the operator L.