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58
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... 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 LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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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 LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Wavelets and the Angle between Past and Future
 Journal of functional analysis
, 1995
"... this paper we consider the weighted norm inequalities with matrix weight. Namely, let W ..."
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Cited by 20 (6 self)
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this paper we consider the weighted norm inequalities with matrix weight. Namely, let W
Regression in Random Design and Warped Wavelets
 BERNOULLI,10
, 2004
"... 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 alg ..."
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Cited by 19 (0 self)
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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.
Maximal operator and weighted norm inequalities for multilinear singular integrals
 Indiana Univ. Math. J
"... Abstract. The maximal operator associated with multilinear CalderónZygmund 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. ..."
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Cited by 16 (9 self)
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Abstract. The maximal operator associated with multilinear CalderónZygmund 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.
Beltrami operators in the plane
 Duke Math. J
"... We determine optimal L pproperties 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)  ≤kw1 − w2 and k is a constant k<1. We also establish the precise invertibility and sp ..."
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Cited by 13 (1 self)
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We determine optimal L pproperties 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)  ≤kw1 − 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.
Prioritised fuzzy constraint satisfaction problems: axioms, instantiation and validation
, 2003
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Extrapolation and sharp norm estimates for classical operators on weighted Lebegue spaces
 Publ. Math
"... 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, the ..."
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Cited by 9 (5 self)
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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.
Stable determination of the surface impedance of an obstacle by far field measurements
, 2005
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Beltrami operators, nonsymmetric elliptic equations and quantitative Jacobian
"... In recent studies on the Gconvergence of Beltrami operators, a number of issues arouse concerning injectivity properties of families of quasiconformal mappings. Bojarski, D’Onofrio, Iwaniec and Sbordone formulated a conjecture based on the existence of a socalled primary pair. Very recently, Bojar ..."
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Cited by 6 (1 self)
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In recent studies on the Gconvergence of Beltrami operators, a number of issues arouse concerning injectivity properties of families of quasiconformal mappings. Bojarski, D’Onofrio, Iwaniec and Sbordone formulated a conjecture based on the existence of a socalled primary pair. Very recently, Bojarski proved the existence of one such pair. We provide a general, constructive, procedure for obtaining a new rich class of such primary pairs. This proof is obtained as a slight adaptation of previous work by the authors concerning the nonvanishing of the Jacobian of pairs of solutions of elliptic equations in divergence form in the plane. It is proven here that the results previously obtained when the coefficient matrix is symmetric also extend to the nonsymmetric case. We also prove a much stronger result giving a quantitative bound for the Jacobian determinant of the socalled periodic σharmonic sense preserving homeomorphisms of C onto itself.
Pointwise multipliers for reverse Hölder spaces
 Studia Mathematica
, 1994
"... We classify weights which map strong reverse Hölder weight classes to weak reverse Hölder weight spaces under pointwise multiplication. 1. ..."
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Cited by 5 (0 self)
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We classify weights which map strong reverse Hölder weight classes to weak reverse Hölder weight spaces under pointwise multiplication. 1.