Revista Matemática Iberoamericana


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Volume 34, Issue 4, 2018, pp. 1541–1561
DOI: 10.4171/rmi/1035

Published online: 2018-12-03

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

Jiao Chen[1] and Guozhen Lu[2]

(1) Beijing Normal University, China and Chongqing Normal University, Chongqing, China
(2) University of Connecticut, Storrs, USA

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: $$T_mf(x_1, x_2)=\frac{1}{(2\pi)^{n_1+n_2}}\int_{\mathbb{R}^n\times \mathbb{R}^m}m(\xi, \eta)\hat{f}(\xi, \eta)e^{2\pi(x_1\xi+x_2\eta)}\,d\xi \,d\eta.$$ 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 $$\sup_{j,k\in \mathbb{Z}}\|m_{j,k}\|_{W^{(s_1,s_2)}}<\infty,$$ 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 $$\|T_m\|_{H^p\rightarrow H^p}\lesssim \sup_{j,k\in \mathbb{Z}}\|m_{j,k}\|_{W^{(s_1,s_2)}}.$$ 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 $m$, $\|T_m\|_{H^p\rightarrow L^p}\!\lesssim\! \sup_{j,k\in \mathbb{Z}}\|m_{j,k}\|_{W^{(s_1,s_2)}}$ and $\|T_m\|_{{\rm CMO}_p\rightarrow {\rm {CMO}}_p}\lesssim \sup_{j,k\in \mathbb{Z}}\|m_{j,k}\|_{W^{(s_1,s_2)}}$ for all $0 < p\le 1$. Moreover, $\|T_m\|_{L^p\rightarrow L^p}\lesssim \sup_{j,k\in \mathbb{Z}}\|m_{j,k}\|_{W^{(s_1,s_2)}}$ for all $1 < p < \infty$ under the assumption $s_1 > n_1/2$ and $s_2 > n_2/2$.

Keywords: Hörmander multiplier, minimal smoothness condition, Littlewood–Paley–Stein square functions, bi-parameter Hardy $H^p$ spaces, bi-parameter Sobolev spaces

Chen Jiao, Lu Guozhen: Hörmander type theorem on bi-parameter Hardy spaces for bi-parameter Fourier multipliers with optimal smoothness. Rev. Mat. Iberoam. 34 (2018), 1541-1561. doi: 10.4171/rmi/1035