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

In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc

Abstract. We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in L2 for small complex L ∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for k-forms are

Abstract. We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with L 2 boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms. MSC classes: 35J25, 35J55, 47N20, 47F05, 42B25 Keywords: Littlewood-Paley estimate, functional calculus, boundary value problems, second order elliptic equations and systems, square root problem 1.

We establish a connection between the absolute continuity of elliptic measure associated to a second order divergence form operator with bounded measurable coefficients with the solvability of an endpoint BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO Dirichlet problem implies L p solvability for all p> p0. 1

Abstract. We consider the Dirichlet problem Lu = 0 in D u = g on ∂D for two second order elliptic operators Lku = ∑n i,j=1 ai,j k (x) ∂iju(x), k = 0, 1, in a bounded Lipschitz domain D ⊂ IR n. The coefficients a i,j k belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that L0 is regular in Lp (∂D, dσ) for some p, 1 < p < ∞, that is, ‖Nu‖Lp ≤ C ‖g‖Lp for all continuous boundary data g. Here σ is the surface measure on ∂D and Nu is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients εi,j (x) = a i,j 1 (x) − ai,j 0 (x) that will assure the perturbed operator L1 to be regular in Lq (∂D, dσ) for some q, 1 < q < ∞. 1.

Abstract. Given two elliptic operators L0 and L1 in nondivergence form, with coefficients Aℓ and drift terms bℓ, ℓ = 0, 1 satisfying sup |Y −X| ≤ δ(X)

Abstract. We consider second order elliptic divergence form equations with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A ∈ L∞(R n;C n+1) for which boundary value problems with L2 Dirichlet or Neumann data are well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. In particular, our methods give a new proof of a theorem of Dahlberg, Jerison and Kenig on the comparability of square functions and non tangential maximal functions of solutions to real symmetric equations. MSC classes: 35J25, 35J55, 47N20 1.

Abstract. Using the framework of a previous article joint with Axelsson and McIntosh, we extend to systems two results of S. Hofmann for real symmetric equations and their perturbations going back to a work of B. Dahlberg for Laplace’s equation on Lipschitz domains, The first one is a certain bilinear estimate for a class of weak solutions and the second is a criterion which allows to identify the domain of the generator of the semi-group yielding such solutions.