Vol. 8, No. 3, 2015

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$L^{p}$ estimates for bilinear and multiparameter Hilbert transforms

Wei Dai and Guozhen Lu

Vol. 8 (2015), No. 3, 675–712
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.

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Keywords
bilinear and multiparameter Hilbert transforms, $L^p$ estimates, Hölder estimates, polydiscs, multiparameter paraproducts, wave packets
Mathematical Subject Classification 2010
Primary: 42B15, 42B20
Milestones
Received: 3 May 2014
Accepted: 22 January 2015
Published: 3 June 2015
Authors
Wei Dai
School of Mathematical Sciences
Beijing Normal University
Beijing, 100875
China
School of Mathematics and Systems Science
Beihang University (BUAA)
Beijing 100191
China
Guozhen Lu
Department of Mathematics
Wayne State University
656 W. Kirby Street
Detroit, MI 48202
United States