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

${T}_{m}\left({f}_{1},{f}_{2}\right)\left(x\right):={\int }_{{ℝ}^{4}}m\left(\xi ,\eta \right){\stackrel{̂}{f}}_{1}\left({\xi }_{1},{\eta }_{1}\right){\stackrel{̂}{f}}_{2}\left({\xi }_{2},{\eta }_{2}\right){e}^{2\pi ix\cdot \left(\left({\xi }_{1},{\eta }_{1}\right)+\left({\xi }_{2},{\eta }_{2}\right)\right)}\phantom{\rule{0.3em}{0ex}}d\xi \phantom{\rule{0.3em}{0ex}}d\eta$

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

$|{\partial }_{\xi }^{\alpha }{\partial }_{\eta }^{\beta }m\left(\xi ,\eta \right)|\lesssim \frac{1}{dist{\left(\xi ,{\Gamma }_{1}\right)}^{|\alpha |}}\cdot \frac{1}{dist{\left(\eta ,{\Gamma }_{2}\right)}^{|\beta |}}$

for sufficiently many multi-indices $\alpha =\left({\alpha }_{1},{\alpha }_{2}\right)$ and $\beta =\left({\beta }_{1},{\beta }_{2}\right)$, ${\Gamma }_{i}$ ($i=1,2$) are subspaces in ${ℝ}^{2}$ and $dim{\Gamma }_{1}=0$, $dim{\Gamma }_{2}=1$. Silva partially answered this question and proved that ${T}_{m}$ maps ${L}^{{p}_{1}}×{L}^{{p}_{2}}\to {L}^{p}$ 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 $\left({p}_{1},{p}_{2},p\right)$ 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 ($d\ge 2$) Hilbert transforms with arbitrary symbols satisfying appropriate decay assumptions and having singularity sets ${\Gamma }_{1},\dots ,{\Gamma }_{d}$ with $dim{\Gamma }_{i}=0$ for $i=1,\dots ,d-1$ and $dim{\Gamma }_{d}=1$. Moreover, we establish the same ${L}^{p}$ estimates as BHT for bilinear and biparameter Fourier multipliers of symbols with $dim{\Gamma }_{1}=dim{\Gamma }_{2}=1$ 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 $dim{\Gamma }_{1}=dim{\Gamma }_{2}=1$ 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