Results 1  10
of
20
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 20 (4 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].
Biparameter paraproducts
"... Abstract. In the first part of the paper we prove a biparameter version of a well known multilinear theorem of Coifman and Meyer. As a consequence, we generalize the KatoPonce inequality in nonlinear PDE. Then, we show that the double bilinear Hilbert transform does not satisfy any L p estimates. ..."
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Cited by 14 (2 self)
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Abstract. In the first part of the paper we prove a biparameter version of a well known multilinear theorem of Coifman and Meyer. As a consequence, we generalize the KatoPonce inequality in nonlinear PDE. Then, we show that the double bilinear Hilbert transform does not satisfy any L p estimates. 1.
A counterexample to a multilinear endpoint question of Christ and Kiselev
, 2001
"... Abstract. Christ and Kiselev [2] have established that the generalized eigenfunctions of onedimensional 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 ..."
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Cited by 13 (7 self)
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Abstract. Christ and Kiselev [2] have established that the generalized eigenfunctions of onedimensional 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) 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.
Local estimates and global continuities in Lebesgue spaces for bilinear operators
, 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 “offdiagonal ” d ..."
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Cited by 9 (4 self)
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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 “offdiagonal ” decay. In addition they allow us to prove global continuities in Lebesgue spaces for bilinear operators with spatial dependent symbol.
L p estimates for the biest I. The Walsh case
"... Abstract. We prove L p estimates (Theorem 1.8) for the Walsh model of the “biest”, a trilinear multiplier with singular symbol. The corresponding estimates for the Fourier model will be obtained in the sequel [15] of this paper. 1. introduction The bilinear Hilbert transform can be written (modulo m ..."
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Cited by 8 (6 self)
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Abstract. We prove L p estimates (Theorem 1.8) for the Walsh model of the “biest”, a trilinear multiplier with singular symbol. The corresponding estimates for the Fourier model will be obtained in the sequel [15] of this paper. 1. introduction The bilinear Hilbert transform can be written (modulo minor modifications) as
The biCarleson operator
, 2005
"... Abstract. We prove L p estimates (Theorem 1.3) for the BiCarleson operator defined below. The methods used are essentially based on the treatment of the Walsh analogue of the operator in the prequel [11] of this paper, but with additional technicalities due to the fact that in the Fourier model one ..."
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Cited by 6 (2 self)
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Abstract. We prove L p estimates (Theorem 1.3) for the BiCarleson operator defined below. The methods used are essentially based on the treatment of the Walsh analogue of the operator in the prequel [11] 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. 1. introduction The maximal Carleson operator is the sublinear operator defined by C(f)(x): = sup N
Vector valued inequalities for families of bilinear Hilbert transforms and applications to biparameter problems
, 2012
"... ar ..."
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Multilinear multipliers associated to simplexes of arbitrary length
, 2007
"... In this article we prove that the n linear operator whose symbol is the characteristic function of the simplex ∆n=ξ1<...<ξn is bounded from L 2 ×... × L 2 into L 2/n, generalizing in this way our previous work on the “biest ” operator [12], [13] (which corresponds to the case n=3) as well a ..."
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Cited by 3 (2 self)
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In this article we prove that the n linear operator whose symbol is the characteristic function of the simplex ∆n=ξ1<...<ξn is bounded from L 2 ×... × L 2 into L 2/n, generalizing in this way our previous work on the “biest ” operator [12], [13] (which corresponds to the case n=3) as well as the LaceyThiele theorem on the bilinear Hilbert transform [7], [8] (which corresponds to the case n=2).