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

  • Jiao Chen

    Beijing Normal University, China and Chongqing Normal University, Chongqing, China
  • Guozhen Lu

    University of Connecticut, Storrs, USA

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 () with optimal smoothness:

One of our results (Theorem 1.7) is the following: assume that is a function on satisfying

with , . Then is bounded from to for all , and

Moreover, the smoothness assumption on and is optimal. Here, , where and are suitable cut-off functions on and , respectively, and is a two-parameter Sobolev space on . We also establish that under the same smoothness assumption on the multiplier , and for all . Moreover, for all under the assumption and .

Cite this article

Jiao Chen, Guozhen Lu, Hörmander type theorem on bi-parameter Hardy spaces for bi-parameter Fourier multipliers with optimal smoothness. Rev. Mat. Iberoam. 34 (2018), no. 4, pp. 1541–1561

DOI 10.4171/RMI/1035