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, ChinaGuozhen Lu
University of Connecticut, Storrs, USA
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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