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# Mathematical Research Letters

## Volume 28 (2021)

### Number 4

### Hörmander Fourier multiplier theorems with optimal regularity in bi-parameter Besov spaces

Pages: 1047 – 1084

DOI: https://dx.doi.org/10.4310/MRL.2021.v28.n4.a4

#### Authors

#### Abstract

The main aim of this paper to establish a bi‑parameter version of a theorem of Baernstein and Sawyer [**1**] on boundedness of Fourier multipliers on one-parameter Hardy spaces $H^p (\mathbb{R}^n)$ which improves an earlier result of Calderón and Torchinsky [**2**].More precisely, we prove the boundedness of the bi‑parameter Fourier multiplier operators on the Lebesgue spaces $L^p (\mathbb{R}^{n_1} \times \mathbb{R}^{n_2}) (1 \lt p \lt \infty)$ and bi‑parameter Hardy spaces $H^p (\mathbb{R}^{n_1} \times \mathbb{R}^{n_2}) (0 \lt p \leq 1)$ with optimal regularity for the multiplier being in the bi‑parameter Besov spaces $B^{(\frac{n_1}{2}, \frac{n_2}{2})}_{2,1} (\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$ and $B^{(s_1,s_2)}_{2,q} (\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$. The Besov regularity assumption is clearly weaker than the assumption of the Sobolev regularity. Thus our results sharpen the known Hörmander multiplier theorem for the bi‑parameter Fourier multipliers using the Sobolev regularity in the same spirit as Baernstein and Sawyer improved the result of Calderón and Torchinsky. Our method is differential from the one used by Baernstein and Sawyer in the one-parameter setting. We employ the bi‑parameter Littlewood–Paley–Stein theory and atomic decomposition for the bi‑parameter Hardy spaces $H^p (\mathbb{R}^{n_1} \times \mathbb{R}^{n_2}) (0 \lt p \leq1)$ to establish our main result (Theorem 1.6). Moreover, the bi‑parameter nature involves much more subtlety in our situation where atoms are supported on arbitrary open sets instead of rectangles.

J.C. was supported partly by NNSF of China (Grant No. 11871126).

G.L. was partly supported by a grant from the Simons Foundation (No. 519099).

Received 18 August 2019

Accepted 24 February 2020

Published 22 November 2021