Results 1 - 10 of 29

Uniform estimates for multi-linear operators with one dimensional modulation symmetry

by Camil Muscalu, Terence Tao, Christoph Thiele , 2001
"... In a previous paper [20] in this series, we gave L p estimates for multi-linear operators given by multipliers which are singular on a non-degenerate 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 ..."
Abstract - Cited by 20 (4 self) - Add to MetaCart
In a previous paper [20] in this series, we gave L p estimates for multi-linear operators given by multipliers which are singular on a non-degenerate 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 non-endpoint uniform estimates for the Bilinear Hilbert transform in [12].
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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.

The disc as a bilinear multiplier

by Loukas Grafakos, Xiaochun Li - 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 ..."
Abstract - Cited by 16 (10 self) - Add to MetaCart
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 vector-valued bounds for families of bilinear Hilbert transforms. 1.
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Nahmod A., Boundedness of bilinear operators with non-smooth symbols

by John E. Gilbert, Andrea R. Nahmod , 2000
"... Abstract. We announce the L p-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. We establish 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 eal ..."
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Abstract. We announce the L p-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. We establish 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.
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ON THE HÖRMANDER CLASSES OF BILINEAR PSEUDODIFFERENTIAL OPERATORS

by Árpád Bényi, Diego Maldonado, Virginia Naibo, Rodolfo H. Torres , 2010
"... Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition a ..."
Abstract - Cited by 10 (4 self) - Add to MetaCart
Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with bilinear Calderón-Zygmund kernels. Some boundedness properties for other classes with estimates in the form of Leibniz’ rule are presented as well.
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Local estimates and global continuities in Lebesgue spaces for bilinear operators

by Frédéric Bernicot , 2008
"... In this paper, we first prove some local estimates for bilinear operators (closely related to the bilinear Hilbert transform and similar singular operators) with truncated symbol. Such estimates, in accordance with the Heisenberg uncertainty principle correspond to a description of “off-diagonal ” d ..."
Abstract - Cited by 9 (4 self) - Add to MetaCart
In this paper, we first prove some local estimates for bilinear operators (closely related to the bilinear Hilbert transform and similar singular operators) with truncated symbol. Such estimates, in accordance with the Heisenberg uncertainty principle correspond to a description of “off-diagonal ” decay. In addition they allow us to prove global continuities in Lebesgue spaces for bilinear operators with spatial dependent symbol.
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ON THE BOUNDEDNESS OF BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES

by Diego Maldonado, Virginia Naibo
"... Abstract. We prove mapping properties of the form T: B ˙ α1,q1 p1 × L p2 → B ˙ α2,q2 p3 and T: B ˙ α1,q1 p1 × ˙ B α2,q2 p2 → Lp3, for certain related indices p1, p2, p3, q1, q2, α1, α2 ∈ R, where T is a bilinear Hörmander-Mihlin multiplier or a molecular paraproduct. Applications to bilinear Little ..."
Abstract - Cited by 8 (3 self) - Add to MetaCart
Abstract. We prove mapping properties of the form T: B ˙ α1,q1 p1 × L p2 → B ˙ α2,q2 p3 and T: B ˙ α1,q1 p1 × ˙ B α2,q2 p2 → Lp3, for certain related indices p1, p2, p3, q1, q2, α1, α2 ∈ R, where T is a bilinear Hörmander-Mihlin multiplier or a molecular paraproduct. Applications to bilinear Littlewood-Paley theory are discussed. 1.
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Bilinear oscillatory integrals and boundedness for new bilinear multipliers

by Frédéric Bernicot, Pierre Germain , 2010
"... ..."
Abstract - Cited by 8 (4 self) - Add to MetaCart
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TRANSFERENCE OF BILINEAR MULTIPLIER OPERATORS On Lorentz Spaces

by Oscar Blasco, Francisco Villarroya , 2007
"... Let m(ξ, η) be a bounded continuous function in IR × IR, 0 < pi, qi < ∞ for i = 1,2 and 0 < p3, q3 ≤ ∞ where 1/p1+1/p2 = 1/p3. It is shown that ..."
Abstract - Cited by 7 (1 self) - Add to MetaCart
Let m(ξ, η) be a bounded continuous function in IR × IR, 0 &lt; pi, qi &lt; ∞ for i = 1,2 and 0 &lt; p3, q3 ≤ ∞ where 1/p1+1/p2 = 1/p3. It is shown that

BILINEAR MULTIPLIERS AND TRANSFERENCE

by Oscar Blasco , 2005
"... We give de Leeuw-type transference theorems for bilinear multipliers. In particular, it is shown that bilinear multipliers arising from regulated functions m(ξ,η)inR × R can be transferred to bilinear multipliers acting on T × T and Z × Z. The results follow from the description of bilinear multipli ..."
Abstract - Cited by 6 (1 self) - Add to MetaCart
We give de Leeuw-type transference theorems for bilinear multipliers. In particular, it is shown that bilinear multipliers arising from regulated functions m(ξ,η)inR × R can be transferred to bilinear multipliers acting on T × T and Z × Z. The results follow from the description of bilinear multipliers on the discrete real line acting on L p-spaces.