Hardy spaces and divergence operators on strongly Lipschitz domains of Rn

Let Ω be a strongly Lipschitz domain of Rn. Consider an elliptic second-order divergence operator L (including a boundary condition on ∂Ω) 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 L1. Under suitable assumptions on L, we identify this maximal Hardy space with H1(Rn) if Ω=Rn, with Hr1(Ω) under the Dirichlet boundary condition, and with Hz1(Ω) under the Neumann boundary condition.