The introduction and development of Hardy and BMO spaces on Euclidean spaces Rn in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations. These spaces were studied extensively in [32], [22], [18], [19], [31] and many others.

Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and

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 L 1. Under suitable assumptions on L, we identify this maximal Hardy space with atomic Hardy spaces, namely with H 1 (R n) if Ω = R n, H 1 r(Ω) under the Dirichlet boundary condition, and H1 z (Ω) under the Neumann boundary condition. In particular, we obtain a new proof of the atomic decomposition for H1 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.

1 Introduction and statement of main results 2

On necessary and sufficient conditions for L p-estimates of Riesz transforms associated to elliptic operators on R n and related estimates