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Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces

  • Jiao Chen , Liang Huang EMAIL logo und Guozhen Lu EMAIL logo
Veröffentlicht/Copyright: 21. Oktober 2021

Abstract

We will establish the boundedness of the Fourier multiplier operator Tmf on multi-parameter Hardy spaces Hp(Rn1××Rnr) (0<p1) when the multiplier 𝑚 is of optimal smoothness in multi-parameter Besov spaces B2,q(s1,,sr)(Rn1××Rnr), where

Tmf(x)=Rn1××Rnrm(ξ)f^(ξ)e2πixξdξ

for xRn1××Rnr. We will show

TmHpHpsupj1,,jrZmj1,,jrB2,q(s1,,sr),

where 0<q< and si>ni(1p-12). Here we have used the notation

mj1,,jr(ξ)=m(2j1ξ1,,2jrξr)ψ(1)(ξ1)ψ(r)(ξr),

and ψ(i)(ξi) is a suitable cut-off function on Rni for 1ir. This multi-parameter Hörmander multiplier theorem is in the spirit of the earlier work of Baernstein and Sawyer in the one-parameter setting and sharpens our recent result of Hörmander multiplier theorem in the bi-parameter case which was established using R. Fefferman’s boundedness criterion. Because R. Fefferman’s boundedness criterion fails in the cases of three or more parameters, it is substantially more difficult to establish such Hörmander multiplier theorems in three or more parameters than in the bi-parameter case. To assume only the optimal smoothness on the multipliers, delicate and hard analysis on the sharp estimates of the square functions on arbitrary atoms are required. Our main theorems give the boundedness on the multi-parameter Hardy spaces under the smoothness assumption of the multipliers in multi-parameter Besov spaces and show the regularity conditions to be sharp.

MSC 2010: 42B15; 42B25

Award Identifier / Grant number: cstc2019jcyj-msxmX0374

Funding statement: The first author was partly supported by the Natural Science Foundation of Chongqing (cstc2019jcyj-msxmX0374). The third author was partly supported by a grant from the Simons Foundation.

  1. Communicated by: Christopher D. Sogge

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Received: 2021-08-05
Published Online: 2021-10-21
Published in Print: 2021-11-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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