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

Abstract

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 $$ T_{m}(f_{1},f_{2})(x):=\int_{\mathbb{R}^{4}}m(\xi,\eta)\hat{f_{1}}(\xi_{1},\eta_{1})\hat{f_{2}}(\xi_{2},\eta_{2})e^{2\pi ix\cdot((\xi_{1},\eta_{1})+(\xi_{2},\eta_{2}))}d\xi d\eta $$ satisfies any $L^p$ estimates, where the symbol $m$ satisfies $$ |\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}m(\xi,\eta)|\lesssim\frac{1}{dist(\xi,\Gamma_{1})^{|\alpha|}}\cdot\frac{1}{dist(\eta,\Gamma_{2})^{|\beta|}} $$ for sufficiently many multi-indices $\alpha=(\alpha_{1},\alpha_{2})$ and $\beta=(\beta_{1},\beta_{2})$, $\Gamma_{i}$ ($i=1,2$) are subspaces in $\mathbb{R}^{2}$ and $dim \, \Gamma_{1}=0, \, dim \, \Gamma_{2}=1$. P. Silva answered partially this question in \cite{S} and proved that $T_{m}$ maps $L^{p_1}\times L^{p_2}\rightarrow L^{p}$ 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).