Hardy spaces adapted to elliptic operators on open sets

Let $L=-\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }\cdot )$ be an elliptic operator defined on an open subset of ${\mathbb{R}}^d$, complemented with mixed boundary conditions. Under suitable assumptions on the operator and the geometry, we derive an atomic characterization (depending only on the boundary conditions) for the Hardy space $H_L^1$ defined using an adapted square function for $L$. This generalizes known results of Auscher and Russ in the case of pure Dirichlet/Neumann boundary conditions on Lipschitz domains. In particular, we develop a connection between the harmonic analysis of $L$ and its underlying geometry.