Lp estimates for bilinear and multiparameter Hilbert transforms

Dai, Wei; Lu, Guozhen

doi:10.2140/apde.2015.8.675

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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

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 $L^{p}$ estimates. They also raised a question asking if a bilinear and biparameter multiplier operator defined by

T_{m} (f_{1}, f_{2}) (x) : = \int_{ℝ^{4}} m (ξ, η) {\hat{f}}_{1} (ξ_{1}, η_{1}) {\hat{f}}_{2} (ξ_{2}, η_{2}) e^{2 π i x \cdot ((ξ_{1}, η_{1}) + (ξ_{2}, η_{2}))} d ξ d η

satisfies any $L^{p}$ estimates, where the symbol $m$ satisfies

| \partial_{ξ}^{α} \partial_{η}^{β} m (ξ, η) | ≲ \frac{1}{dist {(ξ, Γ_{1})}^{| α |}} \cdot \frac{1}{dist {(η, Γ_{2})}^{| β |}}

for sufficiently many multi-indices $α = (α_{1}, α_{2})$ and $β = (β_{1}, β_{2})$ , $Γ_{i}$ ( $i = 1, 2$ ) are subspaces in $ℝ^{2}$ and $dim Γ_{1} = 0$ , $dim Γ_{2} = 1$ . Silva partially answered this question and proved that $T_{m}$ maps $L^{p_{1}} \times L^{p_{2}} \to L^{p}$ boundedly when $\frac{1}{p_{1}} + \frac{1}{p_{2}} = \frac{1}{p}$ with $p_{1}$ , $p_{2} > 1$ , $\frac{1}{p_{1}} + \frac{2}{p_{2}} < 2$ and $\frac{1}{p_{2}} + \frac{2}{p_{1}} < 2$ . One notes that the admissible range here for these tuples $(p_{1}, p_{2}, p)$ is a proper subset of the admissible range of the bilinear Hilbert transform (BHT) derived by Lacey and Thiele.

We establish the same $L^{p}$ estimates as BHT in the full range for the bilinear and $d$ -parameter ( $d \geq 2$ ) Hilbert transforms with arbitrary symbols satisfying appropriate decay assumptions and having singularity sets $Γ_{1}, \dots, Γ_{d}$ with $dim Γ_{i} = 0$ for $i = 1, \dots, d - 1$ and $dim Γ_{d} = 1$ . Moreover, we establish the same $L^{p}$ estimates as BHT for bilinear and biparameter Fourier multipliers of symbols with $dim Γ_{1} = dim Γ_{2} = 1$ and satisfying some appropriate decay estimates. In particular, our results include the $L^{p}$ estimates as BHT in the full range for certain modified bilinear and biparameter Hilbert transforms of tensor-product type with $dim Γ_{1} = dim Γ_{2} = 1$ but with a slightly better logarithmic decay than that of the bilinear and biparameter Hilbert transform $BHT \otimes BHT$ .

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

Vol. 8, No. 3, 2015