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 (q−(L), q+(L)) be the maximal interval of exponents q ∈ [1, ∞] such that the gradient semigroup { √ t∇e}t>0 is bounded on L(R). In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces H L(R) for all p ∈ (0, q+(L)), which, when p = 1, answers a question asked by Deng et al. in [J. Funct. Anal. 263 (2012), 604674]. Moreover, the authors characterize H L(R) via various versions of square functions and Lusin-area functions associated to the operator L.