Results 1 - 10 of 62

Multilinear Calderón Zygmund theory

by Loukas Grafakos, Rodolfo H. Torres - ADV. IN MATH. 40 , 1996
"... A systematic treatment of multilinear Calderón-Zygmundoperators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T1 theorem, anda variety of results regarding multilinear multiplier operators. ..."
Abstract - Cited by 93 (19 self) - Add to MetaCart
A systematic treatment of multilinear Calderón-Zygmundoperators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T1 theorem, anda variety of results regarding multilinear multiplier operators.

Uniform bounds for the bilinear Hilbert transforms

by Xiaochun Li - 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 | ..."
Abstract - Cited by 36 (15 self) - Add to MetaCart
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.

Some remarks on multilinear maps and interpolation

by Loukas Grafakos, Nigel Kalton
"... Abstract. A multilinear version of the Boyd interpolation theorem is proved in the context of quasi-normed rearrangement-invariant spaces. A multilinear Marcinkiewicz interpolation theorem is obtained as a corollary. Several applications are given, including estimates for bilinear fractional integra ..."
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Abstract. A multilinear version of the Boyd interpolation theorem is proved in the context of quasi-normed rearrangement-invariant spaces. A multilinear Marcinkiewicz interpolation theorem is obtained as a corollary. Several applications are given, including estimates for bilinear fractional integrals. 1.
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Bilinear operators with non-smooth symbol

by John E. Gilbert, Andrea R. Nahmod - I, J. Fourier Anal. Appl
"... � � � This paper proves the L p-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. ..."
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� � � This paper proves the L p-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of Coifman-Meyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth. Using a Whitney decomposition in the Fourier plane a general bilinear operator is represented as infinite discrete sums of time-frequency paraproducts obtained by associating wave-packets with tiles in phase-plane. Boundedness for the general bilinear operator then follows once the corresponding L p-boundedness of time-frequency paraproducts has been established. The latter result is the main theorem proved in Part II, our subsequent paper [11], using phase-plane analysis. 1.
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On multilinear singular integrals of Calderón-Zygmund type

by Loukas Grafakos, Rodolfo H. Torres , 2011
"... A variety of results regarding multilinear Calderón-Zygmund singular integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discret ..."
Abstract - Cited by 19 (4 self) - Add to MetaCart
A variety of results regarding multilinear Calderón-Zygmund singular integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur’s test, and a multilinear version of the T1 Theorem suitable for the study of multilinear pseudodifferential and translation invariant operators. A maximal operator associated with multilinear singular integrals is also introduced and employed to obtain weighted norm inequalities.

Breaking duality in the Return Times Theorem

by Ciprian Demeter, Michael Lacey, Terence Tao, Christoph Thiele , 2006
"... 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 Wiener-Win ..."
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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 Wiener-Wintner type of result for the ergodic Hilbert series.

Multilinear interpolation between adjoint operators

by Loukas Grafakos, Terence Tao , 2003
"... Abstract. Multilinear interpolation is a powerful tool used in obtainingstrong type boundedness for a variety of operators assumingonly a finite set of restricted weak-type estimates. A typical situation occurs when one knows that a multilinear operator satisfies a weak L q estimate for a single ind ..."
Abstract - Cited by 17 (9 self) - Add to MetaCart
Abstract. Multilinear interpolation is a powerful tool used in obtainingstrong type boundedness for a variety of operators assumingonly a finite set of restricted weak-type estimates. A typical situation occurs when one knows that a multilinear operator satisfies a weak L q estimate for a single index q (which may be less than one) and that all the adjoints of the multilinear operator are of similar nature, and thus they also satisfy the same weak L q estimate. Under this assumption, in this expository note we give a general multilinear interpolation theorem which allows one to obtain strongtype boundedness for the operator (and all of its adjoints) for a large set of exponents. The key point in the applications we discuss is that the interpolation theorem can handle the case q ≤ 1. When q>1, weak L q has a predual, and such strongtype boundedness can be easily obtained by duality and multilinear interpolation (c.f. [1], [5], [7], [12], [14]). 1. Multilinear operators We begin by setting up some notation for multilinear operators. Let m ≥ 1bean integer. We suppose that for 0 ≤ j ≤ m, (Xj,µj) are measure spaces endowed with
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Transference on certain multilinear multiplier operators

by Dashan Fan, See Profile, Available From Dashan Fan, Dashan Fan, Shuichi Sato, Communicated A. H. Dooley - MR1808390 (2002c:42013
"... multilinear multiplier operators ..."
Abstract - Cited by 14 (0 self) - Add to MetaCart
multilinear multiplier operators

A counterexample to a multilinear endpoint question of Christ and Kiselev

by Camil Muscalu, Terence Tao, Christoph Thiele , 2001
"... Abstract. Christ and Kiselev [2] have established that the generalized eigenfunctions of one-dimensional Dirac operators with L p potential F are bounded for almost all energies for p < 2. Roughly speaking, the proof involved writing these eigenfunctions as a multilinear series ∑ n Tn(F,..., F) a ..."
Abstract - Cited by 13 (7 self) - Add to MetaCart
Abstract. Christ and Kiselev [2] have established that the generalized eigenfunctions of one-dimensional Dirac operators with L p potential F are bounded for almost all energies for p &lt; 2. Roughly speaking, the proof involved writing these eigenfunctions as a multilinear series ∑ n Tn(F,..., F) and carefully bounding each term Tn(F,..., F). It is conjectured that the results in [2] also hold for L 2 potentials F. However in this note we show that the bilinear term T2(F, F) and the trilinear term T3(F, F, F) are badly behaved on L 2, which seems to indicate that multilinear expansions are not the right tool for tackling this endpoint case. 1.
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Pointwise convergence of the ergodic bilinear Hilbert transform, preprint available at http://arxiv.org/abs/math.CA/0601277

by Ciprian Demeter
"... Abstract. Let X = (X, Σ, m, τ) be a dynamical system. We prove that the bilinear series ∑ ′N f(τ n=−N n x)g(τ −n x) n converges almost everywhere for each f, g ∈ L ∞ (X). We also give a proof along the same lines of Bourgain’s analog result for averages. 1. ..."
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Abstract. Let X = (X, Σ, m, τ) be a dynamical system. We prove that the bilinear series ∑ ′N f(τ n=−N n x)g(τ −n x) n converges almost everywhere for each f, g ∈ L ∞ (X). We also give a proof along the same lines of Bourgain’s analog result for averages. 1.
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