Anal. PDE 8 (3), 675-712, (2015) DOI: 10.2140/apde.2015.8.675
KEYWORDS: bilinear and multiparameter Hilbert transforms, $L^P$ estimates, Hölder estimates, polydiscs, multiparameter paraproducts, wave packets, 42B15, 42B20

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

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

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

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

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 ${\mathbb{R}}^{2}$ and $dim{\Gamma}_{1}=0$, $dim{\Gamma}_{2}=1$. Silva partially answered this question and proved that ${T}_{m}$ maps ${L}^{{p}_{1}}\times {L}^{{p}_{2}}\to {L}^{p}$ 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 $\left({p}_{1},{p}_{2},p\right)$ 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 $BHT\otimes BHT$.