Startseite Mathematik Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces
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Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces

  • Liang Huang und Jiao Chen EMAIL logo
Veröffentlicht/Copyright: 5. Februar 2020

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 Calderón–Torchinsky type theorem for the following Fourier multipliers on anisotropic product Hardy spaces Hp(n1×n2;A) (0<p1):

Tmf(x,y)=n1×n2m(ξ,η)f^(ξ,η)e2πi(xξ+yη)dξdη.

Our main theorem is the following: Assume that m(ξ,η) is a function on n1×n2 satisfying

supj,kmj,kW(s1,s2)(A)<

with s1>ζ1,--1(1p-12), s2>ζ2,--1(1p-12), where ζ1,- and ζ2,- depend only on the eigenvalues and are defined in the first section. Then Tm is bounded from Hp(n1×n2;A) to Hp(n1×n2;A) for all 0<p1 and

TmHp(A)Hp(A)CA,s1,s2,psupj,kmj,kW(s1,s2)(A),

where W(s1,s2)(A) is a bi-parameter anisotropic Sobolev space on n1×n2 with CA,s1,s2,p is a positive constant that depends on A,s1,s2,p. Here we use the notations mj,k(ξ,η)=m(A1jξ,A2kη)φ(1)(ξ)φ(2)(η), where φ(1)(ξ) is a suitable cut-off function on n1 and φ(2)(η) is a suitable cut-off function on n2, respectively.

MSC 2010: 42B15; 42B25

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 11801049

Award Identifier / Grant number: cstc2019jcyj-msxmX0374

Funding statement: The authors were supported by NNSF of China (Grant No. 11801049), the Natural Science Foundation of Chongqing (cstc2019jcyj-msxmX0374), Technology Project of Chongqing Education Committee (Grant No. KJQN201800514).

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Received: 2019-09-27
Revised: 2019-12-10
Published Online: 2020-02-05
Published in Print: 2020-05-01

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