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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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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].
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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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.
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.
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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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
GENERALIZATIONS OF THE CARLESONHUNT THEOREM I. THE CLASSICAL SINGULARITY CASE
, 2005
"... Abstract. In this article, we prove L p estimates for a general maximal operator, which extend both the classical CoifmanMeyer [2] and CarlesonHunt [1], [7] theorems in harmonic analysis. 1. ..."
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Abstract. In this article, we prove L p estimates for a general maximal operator, which extend both the classical CoifmanMeyer [2] and CarlesonHunt [1], [7] theorems in harmonic analysis. 1.
On the biCarleson operator. I. The Walsh case
, 2002
"... We prove L p estimates for the Walsh model of the maximal biCarleson operator defined below. The corresponding estimates for the Fourier model will be obtained in the sequel [20] of this paper. ..."
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We prove L p estimates for the Walsh model of the maximal biCarleson operator defined below. The corresponding estimates for the Fourier model will be obtained in the sequel [20] of this paper.