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36
Uniform estimates for the bilinear Hilbert transform
 II, Revista Mat. Iberoamericana
, 2006
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UNIFORM ESTIMATES FOR SOME PARAPRODUCTS
, 2007
"... We establish L p × L q to L r estimates for some general paraproducts, which arise in the study of the bilinear Hilbert transform along curves. ..."
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We establish L p × L q to L r estimates for some general paraproducts, which arise in the study of the bilinear Hilbert transform along curves.
Boundedness of bilinear multipliers whose symbols have a narrow support
 J. Anal. Math
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Multilinear singular operators with fractional rank
 Pacific J. Math
"... Abstract. We prove bounds for multilinear operators on R d given by multipliers which are singular along a k dimensional subspace. The new case of interest is when the rank k/d is not an integer. Connections with the concept of true complexity from Additive Combinatorics are also investigated. ..."
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Abstract. We prove bounds for multilinear operators on R d given by multipliers which are singular along a k dimensional subspace. The new case of interest is when the rank k/d is not an integer. Connections with the concept of true complexity from Additive Combinatorics are also investigated.
Singular integrals meet modulation invariance
 Proceedings ICM 2002 Beijing, Volume II 3 Muscalu
"... Many concepts of Fourier analysis on Euclidean spaces rely on the specification of a frequency point. For example classical Littlewood Paley theory decomposes the spectrum of functions into annuli centered at the origin. In the presence of structures which are invariant under translation of the spec ..."
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Many concepts of Fourier analysis on Euclidean spaces rely on the specification of a frequency point. For example classical Littlewood Paley theory decomposes the spectrum of functions into annuli centered at the origin. In the presence of structures which are invariant under translation of the spectrum (modulation) these concepts need to be refined. This was first done by L. Carleson in his proof of almost everywhere convergence of Fourier series in 1966. The work of M. Lacey and the author in the 1990’s on the bilinear Hilbert transform, a prototype of a modulation invariant singular integral, has revitalized the theme. It is now subject of active research which will be surveyed in the lecture. Most of the recent related work by the author is joint
Lp ESTIMATES FOR A SINGULAR INTEGRAL OPERATOR MOTIVATED BY CALDERÓN’S SECOND COMMUTATOR
, 2012
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METHOD OF ROTATIONS FOR BILINEAR SINGULAR INTEGRALS
"... Abstract. Suppose that Ω lies in the Hardy space H 1 of the unit circle S 1 in R 2. We use the Calderón–Zygmund method of rotations and the uniform boundedness of the bilinear Hilbert transforms to show that the bilinear singular operator with the rough kernel p.v. Ω(x/x)x  −2 is bounded from L ..."
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Abstract. Suppose that Ω lies in the Hardy space H 1 of the unit circle S 1 in R 2. We use the Calderón–Zygmund method of rotations and the uniform boundedness of the bilinear Hilbert transforms to show that the bilinear singular operator with the rough kernel p.v. Ω(x/x)x  −2 is bounded from L p (R) × L q (R) to L r (R), for a large set of indices satisfying 1/p + 1/q = 1/r. We also provide an example of a function Ω in L q (S 1) with mean value zero to show that the singular integral operator given by convolution with p.v. Ω(x/x)x  −2 is not bounded from L p1 p2 p
GENERALIZATIONS OF THE CARLESONHUNT THEOREM I. THE CLASSICAL SINGULARITY CASE
, 2005
"... Abstract. In this article, we prove L p estimates for a general maximal operator, which extend both the classical CoifmanMeyer [2] and CarlesonHunt [1], [7] theorems in harmonic analysis. 1. ..."
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Abstract. In this article, we prove L p estimates for a general maximal operator, which extend both the classical CoifmanMeyer [2] and CarlesonHunt [1], [7] theorems in harmonic analysis. 1.
ON THE HILBERT TRANSFORM AND C 1+ǫ FAMILIES OF LINES
, 2004
"... Abstract. We study the operator ∫ 1 Hvf(x): = p.v. f(x − yv(x)) dy y −1 defined for smooth functions on the plane and measurable vector fields v from the plane into the unit circle. We prove that if v has 1+ǫ derivatives, then Hv extends to a bounded map from L2 (R2) into itself. What is noteworthy ..."
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Abstract. We study the operator ∫ 1 Hvf(x): = p.v. f(x − yv(x)) dy y −1 defined for smooth functions on the plane and measurable vector fields v from the plane into the unit circle. We prove that if v has 1+ǫ derivatives, then Hv extends to a bounded map from L2 (R2) into itself. What is noteworthy is that this result holds in the absence of some additional geometric condition imposed upon v, and that the smoothness condition is nearly optimal. Whereas Hv is a Radon transform for which there is an extensive theory, see e.g. [5], our methods of proof are necessarily those associated to Carleson’s theorem on Fourier series [3]. A previous paper of the authors [10], has shown how to adapt the approach of Lacey and Thiele [11] to the current setting; herein those ideas are combined with a crucial maximal function