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

  • Jiao Chen , Liang Huang EMAIL logo and Guozhen Lu EMAIL logo
Published/Copyright: October 21, 2021
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 6

Abstract

We will establish the boundedness of the Fourier multiplier operator Tm⁢f on multi-parameter Hardy spaces Hp⁢(Rn1×⋯×Rnr) (0<p≤1) when the multiplier 𝑚 is of optimal smoothness in multi-parameter Besov spaces B2,q(s1,…,sr)⁢(Rn1×⋯×Rnr), where

Tm⁢f⁢(x)=∫Rn1×⋯×Rnrm⁢(ξ)⁢f^⁢(ξ)⁢e2⁢π⁢i⁢x⋅ξ⁢dξ

for x∈Rn1×⋯×Rnr. We will show

∥Tm∥Hp→Hp≲supj1,…,jr∈Z⁡∥mj1,…,jr∥B2,q(s1,…,sr),

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

mj1,…,jr⁢(ξ)=m⁢(2j1⁢ξ1,…,2jr⁢ξr)⁢ψ(1)⁢(ξ1)⁢⋯⁢ψ(r)⁢(ξr),

and ψ(i)⁢(ξi) is a suitable cut-off function on Rni for 1≤i≤r. 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.

Keywords: Fourier multipliers; optimal regularity; multi-parameter Besov spaces; multi-parameter Hardy spaces
MSC 2010: 42B15; 42B25

Funding source: Natural Science Foundation of Chongqing

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

Articles in the same Issue

  1. Frontmatter
  2. On the topological complexity of manifolds with abelian fundamental group
  3. Reconstructing étale groupoids from semigroups
  4. Fiberwise linear differential operators
  5. Foliations with isolated singularities on Hirzebruch surfaces
  6. On a stronger reconstruction notion for monoids and clones
  7. Cochain level May–Steenrod operations
  8. Commutative L-algebras and measure theory
  9. Rankin–Selberg integrals for principal series representations of GL(n)
  10. A simple proof of the generalized Leibniz rule on bounded Euclidean domains
  11. Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
  12. Seshadri constants on some Quot schemes
  13. Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
  14. On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
  15. C-minimal topological groups
  16. Cevian properties in ideal lattices of Abelian ℓ-groups
  17. Study of twisted Bargmann transform via Bargmann transform
https://doi.org/10.1515/forum-2021-0201
Keywords for this article
Fourier multipliers; optimal regularity; multi-parameter Besov spaces; multi-parameter Hardy spaces

Articles in the same Issue

  1. Frontmatter
  2. On the topological complexity of manifolds with abelian fundamental group
  3. Reconstructing étale groupoids from semigroups
  4. Fiberwise linear differential operators
  5. Foliations with isolated singularities on Hirzebruch surfaces
  6. On a stronger reconstruction notion for monoids and clones
  7. Cochain level May–Steenrod operations
  8. Commutative L-algebras and measure theory
  9. Rankin–Selberg integrals for principal series representations of GL(n)
  10. A simple proof of the generalized Leibniz rule on bounded Euclidean domains
  11. Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
  12. Seshadri constants on some Quot schemes
  13. Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
  14. On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
  15. C-minimal topological groups
  16. Cevian properties in ideal lattices of Abelian ℓ-groups
  17. Study of twisted Bargmann transform via Bargmann transform
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