Results 1  10
of
47
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 82 (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 65 (20 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 49 (14 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...
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 28 (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 19 (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, preprint
 2001] ESTIMATES ON PARAPRODUCTS 13
"... Abstract. 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, an ..."
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Cited by 16 (5 self)
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Abstract. 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]. 1.
The disc as a bilinear multiplier
 Amer. J. Math
"... Abstract. A classical theorem of C. Fefferman [3] says that the characteristic function of the unit disc is not a Fourier multiplier on L p (R 2) unless p = 2. In this article we obtain a result that brings a contrast with the previous theorem. We show that the characteristic function of the unit di ..."
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Cited by 14 (12 self)
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Abstract. A classical theorem of C. Fefferman [3] says that the characteristic function of the unit disc is not a Fourier multiplier on L p (R 2) unless p = 2. In this article we obtain a result that brings a contrast with the previous theorem. We show that the characteristic function of the unit disc in R 2 is the Fourier multiplier of a bounded bilinear operator from L p1 p2 p p1p2 (R) × L (R) intoL(R), when 2 ≤ p1,p2 < ∞ and 1 <p = ≤ 2. The proof p1+p2 of this result is based on a new decomposition of the unit disc and delicate orthogonality and combinatorial arguments. This result implies norm convergence of bilinear Fourier series and strengthens the uniform boundedness of the bilinear Hilbert transforms, as it yields uniform vectorvalued bounds for families of bilinear Hilbert transforms. 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 12 (9 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.
Density, overcompleteness, and localization of frames, I. Theory
, 2006
"... Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more ..."
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Cited by 12 (3 self)
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Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames F = {fi}i∈I and E = {ej}j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a: I → G. A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of F, the relative measure of E — both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements — and the density of the set a(I) in G. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of
Discrete decompositions for bilinear operators and almost diagonal conditions
 TRANS. AMER. MATH. SOC
, 1998
"... Using discrete decomposition techniques,bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This ..."
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Cited by 10 (6 self)
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Using discrete decomposition techniques,bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This condition should be considered as the direct analogue of an almost diagonal condition for linear operators of CalderónZygmund type. Applications include a reduced T 1 theorem for bilinear pseudodifferential operators and the extension of an L p multiplier result of Coifman and Meyer to the full range of H p spaces. The results of this article rely on decomposition techniques developed by Frazier and Jawerth and on the vector valued maximal estimate of Fefferman and Stein.