2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63237Let $Ω$ be a strongly Lipschitz domain of $\reel^n$. Consider an elliptic second order divergence operator $L$ (including a boundary condition on $\partialΩ$) and define a Hardy space by imposing the non-tangential maximal function of the extension of a function $f$ via the Poisson semigroup for $L$ to be in$L^1$. Under suitable assumptions on $L$, we identify this maximal Hardy space with atomic Hardy spaces, namely with $H^1(\reel^n)$ if $Ω=\reel^n$, $H^{1}_{r}(Ω)$ under the Dirichlet boundary condition, and $H^{1}_{z}(Ω)$ under the Neumann boundary condition. In particular, we obtain a new proof of the atomic decomposition for $H^{1}_{z}(Ω)$. A version for local Hardy spaces is also given. We also present an overview of the theory of Hardy spaces and BMO spaces on Lipschitz domains with proofs.submittedClassical Analysis and ODEs42B30, 42B25text