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93
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 53 (5 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.
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 31 (12 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.
Bilinear operators with nonsmooth symbol
 I, J. Fourier Anal. Appl
"... � � � This paper proves the L pboundedness 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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Cited by 29 (3 self)
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� � � This paper proves the L pboundedness 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 CoifmanMeyer for smooth multipliers and ones, such the Bilinear Hilbert transform of LaceyThiele, 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 timefrequency paraproducts obtained by associating wavepackets with tiles in phaseplane. Boundedness for the general bilinear operator then follows once the corresponding L pboundedness of timefrequency paraproducts has been established. The latter result is the main theorem proved in Part II, our subsequent paper [11], using phaseplane analysis. 1.
The Marcinkiewicz multiplier condition for bilinear operators
 Studia Math. 146 (2001), 115–156. LOUKAS GRAFAKOS
"... Abstract. This article is concerned with the question of whether Marcinkiewicz multipliers on R2n give rise to bilinear multipliers on Rn × Rn.We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions ..."
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Cited by 28 (8 self)
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Abstract. This article is concerned with the question of whether Marcinkiewicz multipliers on R2n give rise to bilinear multipliers on Rn × Rn.We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces. 1.
Geometric renormalization of large energy wave maps
, 2004
"... There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for twodimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or ..."
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Cited by 24 (7 self)
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There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for twodimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the solution is large. In this report we present a different renormalization, based on the harmonic map heat flow, which works for large energy wave maps from two dimensions to hyperbolic spaces. We also observe an intriguing estimate of “nonconcentration” type, which asserts roughly speaking that if the energy of a wave map concentrates at a point, then it becomes asymptotically selfsimilar.
On multilinear singular integrals of CalderónZygmund type
, 2011
"... A variety of results regarding multilinear CalderónZygmund 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 ..."
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Cited by 19 (4 self)
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A variety of results regarding multilinear CalderónZygmund 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.
Modulation spaces and a class of bounded multilinear pseudodifferential operators
 J. OPERATOR THEORY
, 2003
"... We show that multilinear pseudodifferential operators with symbols in the modulation space M ∞,1 are bounded on products of modulation spaces. In particular, M ∞,1 includes nonsmooth symbols. Several multilinear Calderón– Vaillancourttype theorems are then obtained by using certain embeddings of c ..."
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Cited by 17 (8 self)
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We show that multilinear pseudodifferential operators with symbols in the modulation space M ∞,1 are bounded on products of modulation spaces. In particular, M ∞,1 includes nonsmooth symbols. Several multilinear Calderón– Vaillancourttype theorems are then obtained by using certain embeddings of classical function spaces into modulation spaces.
Multilinear interpolation between adjoint operators
, 2003
"... Abstract. Multilinear interpolation is a powerful tool used in obtainingstrong type boundedness for a variety of operators assumingonly a finite set of restricted weaktype estimates. A typical situation occurs when one knows that a multilinear operator satisfies a weak L q estimate for a single ind ..."
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Cited by 17 (9 self)
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Abstract. Multilinear interpolation is a powerful tool used in obtainingstrong type boundedness for a variety of operators assumingonly a finite set of restricted weaktype 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
Multilinear CalderonZygmund operators on Hardy spaces
 Collect. Math
"... Abstract. It is shown that multilinear CalderónZygmund operators are bounded on products of Hardy spaces. 1. ..."
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Abstract. It is shown that multilinear CalderónZygmund operators are bounded on products of Hardy spaces. 1.