Hörmander type theorem on bi-parameter Hardy spaces for bi-parameter Fourier multipliers with optimal smoothness

Abstract

The main purpose of this paper is to establish, using the bi-parameter Littlewood–Paley–Stein theory (in particular, the bi-parameter Littlewood–Paley–Stein square functions), a Hörmander–Mihlin type theorem for the following bi-parameter Fourier multipliers on bi-parameter Hardy spaces $H^p({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$ ($0<p\le 1$) with optimal smoothness:

One of our results (Theorem 1.7) is the following: assume that $m(\xi ,\eta )$ is a function on ${\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_1}$ satisfying

with $s_1>n_1(1/p-1/2)$, $s_2>n_2(1/p-1/2)$. Then $T_m$ is bounded from $H^p({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$ to $H^p({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$ for all $0<p\le 1$, and

Moreover, the smoothness assumption on $s_1$ and $s_2$ is optimal. Here, $m_{j,k}(\xi ,\eta )=m(2^j\xi ,2^k\eta )\Psi (\xi )\Psi (\eta )$, where $\Psi (\xi )$ and $\Psi (\eta )$ are suitable cut-off functions on ${\mathbb{R}}^{n_1}$ and ${\mathbb{R}}^{n_2}$, respectively, and $W^{(s_1,s_2)}$ is a two-parameter Sobolev space on ${\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}$. We also establish that under the same smoothness assumption on the multiplier $m$, $\Vert T_m{\Vert }_{H^p\to L^p}\lesssim {\sup }_{j,k\in \mathbb{Z}}\Vert m_{j,k}{\Vert }_{W^{(s_1,s_2)}}$ and $\Vert T_m{\Vert }_{{\mathrm{CMO}}_p\to {\mathrm{CMO}}_p}\lesssim {\sup }_{j,k\in \mathbb{Z}}\Vert m_{j,k}{\Vert }_{W^{(s_1,s_2)}}$ for all $0<p\le 1$. Moreover, $\Vert T_m{\Vert }_{L^p\to L^p}\lesssim {\sup }_{j,k\in \mathbb{Z}}\Vert m_{j,k}{\Vert }_{W^{(s_1,s_2)}}$ for all $1<p<\infty$ under the assumption $s_1>n_1/2$ and $s_2>n_2/2$.