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

Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L) −1/2 on the L p space. As an application, for 1 < p < 3 + ε, we establish the L p boundedness of Riesz transforms on Lipschitz domains for operators with V MO coefficients. The range of p is sharp. The closely related boundedness of ∇(L) −1/2 on weighted L 2 spaces is also studied. 1.

Abstract. Let Ω be a bounded Lipschitz domain in Rn. We develop a new approach to the invertibility on Lp (∂Ω) of the layer potentials associated with elliptic equations and systems in Ω. As a consequence, for n ≥ 4 and 2(n−1) − ε < p < 2 where ε> 0 depends on Ω, n+1 we obtain the solvability of the Lp Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established.

Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(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)

Abstract. Let X be a metric space with doubling measure, and L be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characterization, duality of Hardy and BMO spaces. Further specializing to the case that L is a Schrödinger operator on R n with a non-negative, locally integrable potential, we establish addition characterizations of such Hardy space in terms of maximal functions. Finally, (X) for p> 1, which may or may not coincide with the space L p (X), and show that they interpolate with H 1 L(X) spaces by the complex method. we define Hardy spaces H p L The authors gratefully acknowledge support from NSF as follows: S. Hofmann (DMS

1 Introduction and statement of main results 2

Abstract. We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-λ method that does not use any size or smoothness estimates for the kernels. 1.

Abstract. Let L = divA ∇ be a real, symmetric second order elliptic operator with bounded measurable coefficients. Consider the elliptic equation Lu = 0 in a bounded Lipschitz domain Ω of R n. We study the relationship between the solvability of the L p Dirichlet problem (D)p with boundary data in L p (∂Ω) and that of the L q regularity problem (R)q with boundary data in W 1,q (∂Ω), where 1 < p, q < ∞. It is known that the solvability of (R)p implies that of (D) p ′. In this note we show that if (D) p ′ is solvable, then either (R)p is solvable or (R)q is not solvable for any 1 < q < ∞. 1.

∇ , ε> 0 be a family of second order elliptic operators

On considère la classe des variétés riemanniennes complètes non compactes dont le noyau de la chaleur satisfait une estimation supérieure et inférieure gaussienne. On montre que la transformée de Riesz y est bornée sur L p, pour un intervalle ouvert de p au-dessus de 2, si et seulement si le gradient du noyau de la chaleur satisfait une certaine estimation L p pour le même intervalle d’exposants p. One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same