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

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

$$T_m\left(f_1,f_2\right)\left(x\right):={\int }_{{ℝ}^4}m\left(\xi ,\eta \right){\widehat{f}}_1\left({\xi }_1,{\eta }_1\right){\widehat{f}}_2\left({\xi }_2,{\eta }_2\right)e^{2\pi ix\cdot \left(\left({\xi }_1,{\eta }_1\right)+\left({\xi }_2,{\eta }_2\right)\right)}d\xi d\eta$$

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 ${ℝ}^2$ 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 $BHT\otimes BHT$.