Abstract. Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H p -boundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ∞ functional calculus and Hodge decomposition, are given.

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 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.

We prove that on any connected unimodular Lie group G , the space L p α (G) ∩ L ∞ (G), where L p α (G) is the Sobolev space of order α > 0 associated with a sublaplacian, is an algebra under pointwise product. This generalizes results due to Strichartz (in the Euclidean case), to Bohnke (in the case of stratified groups), and others. A global version of this fact holds for groups with polynomial growth. We give similar results for Riemannian manifolds with Ricci curvature bounded from below, respectively nonnegative.

Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by P f (x) = y p(x, y)f (y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ∇f p and (I − P ) 1/2 f p uniformly in f for 1 < p < +∞. These conditions are different for p < 2 and p > 2. The proofs rely on recent techniques developed to handle operators beyond the class of Calderón-Zygmund operators. For our purpose, we also prove Littlewood-Paley inequalities and interpolation results for Sobolev spaces in this context, which are of independent interest.