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

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.

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.

Abstract. The theory of Carleson measures, stopping time arguments, and atomic decompositions has been well-established in harmonic analysis. More recent is the theory of phase space analysis from the point of view of wave packets on tiles, tree selection algorithms, and tree size estimates. The purpose of this paper is to demonstrate that the two theories are in fact closely related, by taking existing results and reproving them in a unified setting. In particular we give a dyadic version of extrapolation for Carleson measures, as well as a two-sided local dyadic T(b) theorem which generalizes earlier T(b) theorems of David, Journe, Semmes, and Christ.

Abstract. We consider divergence form elliptic operators of the form L= − div A(x)∇, defined inR n+1 ={(x, t)∈R n ×R}, n≥2, where the L ∞ coefficient matrix A is (n+ 1)×(n+1), uniformly elliptic, complex and t-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on L2 (Rn)=L 2 (∂Rn+1 +), is stable under complex, L ∞ perturbations of the coefficient matrix. Using a variant of the Tb Theorem, we also prove that the layer potentials are bounded and invertible on L2 (Rn) whenever A(x) is real and symmetric (and thus, by our stability result, also when A is complex,‖A − A0‖ ∞ is small enough and A0 is real, symmetric, L ∞ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with L2 (resp. ˙L 2 1) data, for small complex perturbations of a real symmetric matrix. Previously, L2 solvability results for complex (or even real but non-symmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients A j,n+1 = 0=An+1, j, 1 ≤ j≤n, which corresponds to the Kato square root problem.

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1 Introduction and statement of main results 2

Abstract. Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMOL space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMOL spaces in the theory of singular integration such as BMOL estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space BMOL might coincide with or might be essentially different from the classical BMO space. 1.

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.