Results 1  10
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93
On Calderón’s conjecture
, 1999
"... This paper is a successor of [4]. In that paper we considered bilinear operators of the form (1) Hα(f1,f2)(x): = p.v. f1(x − t)f2(x + αt) dt ..."
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Cited by 118 (25 self)
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This paper is a successor of [4]. In that paper we considered bilinear operators of the form (1) Hα(f1,f2)(x): = p.v. f1(x − t)f2(x + αt) dt
Multilinear operators given by singular multipliers
 J. Amer. Math. Soc
"... Abstract. We prove L p estimates for a large class of multilinear operators, which includes the multilinear paraproducts studied by Coifman and Meyer [7], as well as the bilinear Hilbert transform. 1. ..."
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Cited by 90 (23 self)
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Abstract. We prove L p estimates for a large class of multilinear operators, which includes the multilinear paraproducts studied by Coifman and Meyer [7], as well as the bilinear Hilbert transform. 1.
L p estimates for the bilinear Hilbert transform
, 1996
"... For the bilinear Hilbert transform given by H fg(x) = p. v. Z f(x \Gamma y)g(x + y) dy y we announce the inequality kH fgk p3 K p1 ;p 2 kfk p1 kgk p2 , provided 2 ! p 1 ; p 2 ! 1, 1=p 3 = 1=p 1 + 1=p 2 and 1 ! p 3 ! 2. We announce a partial resolution to long standing conjectures concerning th ..."
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Cited by 64 (16 self)
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For the bilinear Hilbert transform given by H fg(x) = p. v. Z f(x \Gamma y)g(x + y) dy y we announce the inequality kH fgk p3 K p1 ;p 2 kfk p1 kgk p2 , provided 2 ! p 1 ; p 2 ! 1, 1=p 3 = 1=p 1 + 1=p 2 and 1 ! p 3 ! 2. We announce a partial resolution to long standing conjectures concerning the operator known as the bilinear Hilbert transform, defined as follows H fg(x) = lim ffl!0 Z jyj?ffl f(x \Gamma y)g(x + y) dy y : This operation is initially defined only for certain functions f and g, for instance those in the Schwartz class on R. The conjectures concern the extension of H to a bounded operator on L p spaces. We have proved Theorem 1 H extends to a bounded operator on L p1 \Theta L p2 into L p3 , provided 2 ! p 1 ; p 2 ! 1 and 1 ! p 3 ! 2, where 1=p 3 = 1=p 1 + 1=p 2 . Some thirty years ago, in connection with the Cauchy integral on Lipschitz curves [1] A. P. Calder'on raised the question of H mapping L 2 \Theta L 2 into L 1 . This inequality is true an...
Multilinear weighted convolution of L2 functions, and applications to nonlinear dispersive equation
 Amer. J. Math
"... Multilinear weighted convolution of L 2 functions, and applications to nonlinear dispersive equations ..."
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Cited by 52 (3 self)
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Multilinear weighted convolution of L 2 functions, and applications to nonlinear dispersive equations
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.
Uniform bounds for the bilinear Hilbert transforms
 889–993. MR2113017 (2006e:42011), Zbl 1071.44004. Xiaochun Li
, 2004
"... Abstract. We continue the investigation initiated in [8] of uniform Lp bounds � for the family of bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. f(x − αt)g(x − βt) R dt t. In this work we show that Hα,β map Lp1 (R) × Lp2 (R) into Lp (R) uniformly in the real parameters α, β satisfying  α β − 1  ..."
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Cited by 36 (15 self)
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Abstract. We continue the investigation initiated in [8] of uniform Lp bounds � for the family of bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. f(x − αt)g(x − βt) R dt t. In this work we show that Hα,β map Lp1 (R) × Lp2 (R) into Lp (R) uniformly in the real parameters α, β satisfying  α β − 1  ≥ c> 0 when 1 < p1, p2 < 2 and 2 p1p2 3 < p = < ∞. p1+p2 As a corollary we obtain Lp × L ∞ → Lp uniform bounds in the range 4/3 < p < 4 for the H1,α’s when α ∈ [0, 1). 1.
The dichotomy between structure and randomness, arithmetic progressions, and the primes
"... Abstract. A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness ..."
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Cited by 27 (1 self)
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Abstract. A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (lowcomplexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the GreenTao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different. 1.
Uniform estimates for multilinear operators with one dimensional modulation symmetry
, 2001
"... In a previous paper [20] in this series, we gave L p estimates for multilinear operators given by multipliers which are singular on a nondegenerate subspace of some dimension k. In this paper we give uniform estimates when the subspace approaches a degenerate region in the case k = 1, and when al ..."
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Cited by 21 (5 self)
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In a previous paper [20] in this series, we gave L p estimates for multilinear operators given by multipliers which are singular on a nondegenerate subspace of some dimension k. In this paper we give uniform estimates when the subspace approaches a degenerate region in the case k = 1, and when all the exponents p are between 2 and ∞. In particular we recover the nonendpoint uniform estimates for the Bilinear Hilbert transform in [12].
Thiele Breaking duality in the Return Times Theorem
 Duke Math. J
"... Abstract. We prove Bourgain’s Return Times Theorem for a range of exponents p and q that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence for the signed average analog of Bourgain’s averages. As an immediate corollary we obtain a Wi ..."
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Cited by 20 (10 self)
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Abstract. We prove Bourgain’s Return Times Theorem for a range of exponents p and q that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence for the signed average analog of Bourgain’s averages. As an immediate corollary we obtain a WienerWintner type of result for the ergodic Hilbert series. 1.
L p estimates for the biest II. The Fourier case
"... Abstract. We prove L p estimates (Theorem 1.2) for the “biest”, a trilinear multiplier operator with singular symbol. The methods used are based on the treatment of the Walsh analogue of the biest in the prequel [16] of this paper, but with additional technicalities due to the fact that in the Fouri ..."
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Cited by 20 (10 self)
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Abstract. We prove L p estimates (Theorem 1.2) for the “biest”, a trilinear multiplier operator with singular symbol. The methods used are based on the treatment of the Walsh analogue of the biest in the prequel [16] of this paper, but with additional technicalities due to the fact that in the Fourier model one cannot obtain perfect localization in both space and frequency.