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172
The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical Ap characteristic
 Amer. J. Math
"... Abstract. We show that the norm of the Hilbert transform as an operator in the weighted space Lp R (ω) for 2 ≤ p < ∞ is bounded by a constant multiple of the first power of the classical Ap characteristic of ω. This result is sharp. We also prove a bilinear imbedding theorem with simple condition ..."
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Cited by 64 (4 self)
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Abstract. We show that the norm of the Hilbert transform as an operator in the weighted space Lp R (ω) for 2 ≤ p < ∞ is bounded by a constant multiple of the first power of the classical Ap characteristic of ω. This result is sharp. We also prove a bilinear imbedding theorem with simple conditions.
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 61 (15 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
New maximal functions and multiple weights for the multilinear CalderónZygmund theory
 MATH
, 2010
"... A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller that the mfold product of the HardyLittlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of CalderónZygmund type and to ..."
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Cited by 48 (4 self)
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A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller that the mfold product of the HardyLittlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of CalderónZygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear CalderónZygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp endpoint estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.
The boundedness of classical operators on variable Lp spaces
"... Abstract. We show that many classical operators in harmonic analysis —such as maximal operators, singular integrals, commutators and fractional integrals— are bounded on the variable Lebesgue space Lp(·) whenever the HardyLittlewood maximal operator is bounded on Lp(·). Further, we show that such o ..."
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Cited by 41 (2 self)
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Abstract. We show that many classical operators in harmonic analysis —such as maximal operators, singular integrals, commutators and fractional integrals— are bounded on the variable Lebesgue space Lp(·) whenever the HardyLittlewood maximal operator is bounded on Lp(·). Further, we show that such operators satisfy vectorvalued inequalities. We do so by applying the theory of weighted norm inequalities and extrapolation. As applications we prove the CalderónZygmund inequality for solutions of4u = f in variable Lebesgue spaces, and prove the Calderón extension theorem for variable Sobolev spaces.
Wavelets and the angle between past and future
 DAMIR Z. AROV DEPARTMENT OF MATHEMATICAL ANALYSIS SOUTH UKRANIAN PEDAGOGICAL UNIVERSITY 65020 ODESSA UKRAINE HARRY DYM DEPARTMENT OF MATHEMATICS THE WEIZMANN INSTITUTE OF SCIENCE REHOVOT 76100 ISRAEL dym@wisdom.weizmann.ac.il
, 1997
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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 ..."
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Cited by 35 (7 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.
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 ..."
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Cited by 34 (9 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.
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 28 (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 28 (11 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.
The Schrödinger operator on the energy space: Boundedness and compactness criteria
 Acta Math
"... We characterize the class of measurable functions (or, more generally, real or complexvalued distributions) V such that the Schrödinger operator H = −∆+ V ..."
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Cited by 26 (8 self)
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We characterize the class of measurable functions (or, more generally, real or complexvalued distributions) V such that the Schrödinger operator H = −∆+ V