A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates

Let $L$ be a nonnegative, self-adjoint operator satisfying Gaussian estimates on $L^2({\text{RR}}^n)$. In this article we give an atomic decomposition for the Hardy spaces $H_{L,max}^p(\mathbb{R})$ in terms of the nontangential maximal functions associated with the heat semigroup of $L$, and this leads eventually to characterizations of Hardy spaces associated to $L$, via atomic decomposition or the nontangential maximal functions. The proofs are based on a modification of technique due to A. Calder'on \cite{C}.