$L^{p}$ 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 $T_{m}(f_{1},f_{2})(x):=\int_{\mathbb{R}^{4}}m(ξ,η)\hat{f_{1}}(ξ_{1},η_{1})\hat{f_{2}}(ξ_{2},η_{2})e^{2πix\cdot((ξ_{1},η_{1})+(ξ_{2},η_{2}))}dξdη$ satisfies any $L^p$ estimates, where the symbol $m$ satisfies $|\partial_ξ^α\partial_η^βm(ξ,η)|\lesssim\frac{1}{dist(ξ,Γ_{1})^{|α|}}\cdot\frac{1}{dist(η,Γ_{2})^{|β|}}$ for sufficiently many multi-indices $α=(α_{1},α_{2})$ and $β=(β_{1},β_{2})$, $Γ_{i}$ ($i=1,2$) are subspaces in $\mathbb{R}^{2}$ and $dim \, Γ_{1}=0, \, dim \, Γ_{2}=1$. 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 $dim \, Γ_{1}=dim \, Γ_{2}=1$ but with a slightly better decay than that for the bilinear and bi-parameter Hilbert transform (Theorem 1.4).

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