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 ${q\in[1,\infty]}$ such that the semigroup ${\{e^{-tL}\}_{t>0}}$ is bounded on ${L^{q}(\mathbb{R}^{n})}$. In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces ${H_{L}^{p}(\mathbb{R}^{n})}$ for all ${p\in(0,p_{+}(L))}$, which when ${p=1}$, answers a question asked by Deng, Ding and Yao in [21]. Moreover, the authors characterize ${H_{L}^{p}(\mathbb{R}^{n})}$ via various versions of square functions and Lusin-area functions associated to the operator L.