this paper we consider the weighted norm inequalities with matrix weight. Namely, let W

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. The maximal operator associated with multilinear Calderón-Zygmund singular integrals is introduced and shown to be bounded on product of Lebesgue spaces. Moreover weighted norm inequalities are obtained for this maximal operator as well as for the corresponding singular integrals. 1.

We consider the problem of estimating an unknown function f in a regression setting with random design. Instead of expanding the function on a regular wavelet basis, we expand it on the basis jk (G), j, k} warped with the design. This allows to perform a very stable and computable thresholding algorithm. We investigate the properties of this new basis. In particular, we prove that if the design has a property of Muckenhoupt type, this new basis has a behavior quite similar to a regular wavelet basis. This enables us to prove that the associated thresholding procedure achieves rates of convergence which have been proved to be minimax in the uniform design case.

We determine optimal L p-properties for the solutions of the general nonlinear elliptic system in the plane of the form fz = H(z,fz), h ∈ L p (C), where H is a measurable function satisfying |H(z,w1) − H(z,w2) | ≤k|w1 − w2| and k is a constant k<1. We also establish the precise invertibility and spectral properties in Lp (C) for the operators I − Tµ, I − µT, and T − µ, where T is the Beurling transform. These operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations (PDEs)in two dimensions. In particular, we prove invertibility in Lp (C) whenever 1 +‖µ‖ ∞ <p<1 + 1/‖µ‖∞. We also prove related results with applications to the regularity of weakly quasiconformal mappings. 1.

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We obtain sharp weighted Lp estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 < r < ∞ the norm of a sublinear operator on Lr (w) is bounded by a function of the Ar characteristic constant of the weight w, then for p> r it is bounded on Lp (v) by the same increasing function of the Ap characteristic constant of v, and for p < r it is bounded on Lp (v) by the same increasing function of the r−1 p−1 power of the Ap characteristic constant of v. For some operators these bounds are sharp, but not always. In particular, we show that they are sharp for the Hilbert, Beurling, and martingale transforms.

We classify weights which map strong reverse Hölder weight classes to weak reverse Hölder weight spaces under pointwise multiplication. 1.

A conformal map on the unit disk is called a compact deformation of a Fuchsian group G if has a quasiconformal extension to the plane, h, which conjugates G to a Kleinian group G 0 and the dilatation of h is compactly supported modulo G. We show that for such deformations = dim( ) = dim( c ), and the image of e = n c is contained in a countable union of rectifiable curves and has zero length i G is divergence type.

Abstract. We give new elementary proofs of theorems due to B. Muckenhoupt, B. Jawerth, and S. Buckley. By means of our approach we answer a question raised by J. Orobitg and C. Pérez. 1.