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2015 $L^{p}$ estimates for bilinear and multiparameter Hilbert transforms
Wei Dai, Guozhen Lu
Anal. PDE 8(3): 675-712 (2015). DOI: 10.2140/apde.2015.8.675

Abstract

Muscalu, Pipher, Tao and Thiele proved that the standard bilinear and biparameter Hilbert transform does not satisfy any Lp estimates. They also raised a question asking if a bilinear and biparameter multiplier operator defined by

Tm(f1,f2)(x) :=4m(ξ,η)f̂1(ξ1,η1)f̂2(ξ2,η2)e2πix((ξ1,η1)+(ξ2,η2))dξdη

satisfies any Lp estimates, where the symbol m satisfies

|ξα ηβm(ξ,η)| 1 dist(ξ,Γ1)|α| 1 dist(η,Γ2)|β|

for sufficiently many multi-indices α = (α1,α2) and β = (β1,β2), Γi (i = 1,2) are subspaces in 2 and dimΓ1 = 0, dimΓ2 = 1. Silva partially answered this question and proved that Tm maps Lp1 × Lp2 Lp boundedly when 1 p1 + 1 p2 = 1 p with p1, p2 > 1, 1 p1 + 2 p2 < 2 and 1 p2 + 2 p1 < 2. One notes that the admissible range here for these tuples (p1,p2,p) is a proper subset of the admissible range of the bilinear Hilbert transform (BHT) derived by Lacey and Thiele.

We establish the same Lp estimates as BHT in the full range for the bilinear and d-parameter (d 2) Hilbert transforms with arbitrary symbols satisfying appropriate decay assumptions and having singularity sets Γ1,,Γd with dimΓi = 0 for i = 1,,d 1 and dimΓd = 1. Moreover, we establish the same Lp estimates as BHT for bilinear and biparameter Fourier multipliers of symbols with dimΓ1 = dimΓ2 = 1 and satisfying some appropriate decay estimates. In particular, our results include the Lp estimates as BHT in the full range for certain modified bilinear and biparameter Hilbert transforms of tensor-product type with dimΓ1 = dimΓ2 = 1 but with a slightly better logarithmic decay than that of the bilinear and biparameter Hilbert transform BHTBHT.

Citation

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Wei Dai. Guozhen Lu. "$L^{p}$ estimates for bilinear and multiparameter Hilbert transforms." Anal. PDE 8 (3) 675 - 712, 2015. https://doi.org/10.2140/apde.2015.8.675

Information

Received: 3 May 2014; Accepted: 22 January 2015; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1329.42008
MathSciNet: MR3353828
Digital Object Identifier: 10.2140/apde.2015.8.675

Subjects:
Primary: 42B15, 42B20

Rights: Copyright © 2015 Mathematical Sciences Publishers

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