Hardy spaces and divergence operators on strongly Lipschitz domains of Rn

Abstract Let Ω be a strongly Lipschitz domain of R n . 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 H 1 ( R n ) if Ω= R n , with H r 1 (Ω) under the Dirichlet boundary condition, and with H z 1 (Ω) under the Neumann boundary condition.