$L^p$ estimates for bilinear and multi-parameter Hilbert transforms

C. Muscalu, J. Pipher, T. Tao and C. Thiele proved in \cite{MPTT1} that the standard bilinear and bi-parameter Hilbert transform does not satisfy any $L^p$ estimates. They also raised a question asking if a bilinear and bi-parameter multiplier operator defined by $$T_m(f_1,f_2)(x):={\int }_{{\mathbb{R}}^4}m(\xi ,\eta )\widehat{f_1}({\xi }_1,{\eta }_1)\widehat{f_2}({\xi }_2,{\eta }_2)e^{2\pi ix\cdot (({\xi }_1,{\eta }_1)+({\xi }_2,{\eta }_2))}d\xi d\eta$$ satisfies any $L^p$ estimates, where the symbol $m$ satisfies $$\mathrm{\mid }{\mathrm{\partial }}_{\xi }^{\alpha }{\mathrm{\partial }}_{\eta }^{\beta }m(\xi ,\eta )\mathrm{\mid }\lesssim \frac{1}{dist(\xi ,{\mathrm{\Gamma }}_1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}}\cdot \frac{1}{dist(\eta ,{\mathrm{\Gamma }}_2{)}^{\mathrm{\mid }\beta \mathrm{\mid }}}$$ for sufficiently many multi-indices $\alpha =({\alpha }_1,{\alpha }_2)$ and $\beta =({\beta }_1,{\beta }_2)$, ${\mathrm{\Gamma }}_i$ ($i=1,2$) are subspaces in ${\mathbb{R}}^2$ and $dim{\mathrm{\Gamma }}_1=0,dim{\mathrm{\Gamma }}_2=1$. P. Silva answered partially this question in \cite{S} 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 observes that the admissible range here for these tuples $(p_1,p_2,p)$ is a proper subset contained in the admissible range of BHT. In this paper, we establish the same $L^p$ estimates as BHT in the full range for the bilinear and multi-parameter Hilbert transforms with arbitrary symbols satisfying appropriate decay assumptions (Theorem 1.3). Moreover, we also establish the same $L^p$ estimates as BHT for certain modified bilinear and bi-parameter Hilbert transforms with $dim , \Gamma_{1}=dim , \Gamma_{2}=1$ but with a slightly better decay than that for the bilinear and bi-parameter Hilbert transform (Theorem 1.4).