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 $L^2(\RR^n)$. In this article we give an atomic decomposition for the Hardy spaces $H^p_{L,max}(\R)$ 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}.