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Forum Mathematicum

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Volume 27, Issue 5

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Hardy spaces associated with a pair of commuting operators

Jun Cao
  • School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China
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/ Zunwei Fu / Renjin Jiang
  • School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China
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/ Dachun Yang
  • School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China
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Published Online: 2013-12-05 | DOI: https://doi.org/10.1515/forum-2013-0103

Abstract

Let L1,L2 be a pair of one-to-one commuting sectorial operators such that each Li for i{1,2} satisfies the mi order L2 off-diagonal estimates and m1m2>0. Let HLip(n), i{1,2}, and HL1+L˜2p(n) be the Hardy spaces associated, respectively, to the operators Li and L1+L˜2, where L˜2:=L2m1/m2 is a fractional power of L2. In this paper, the authors give out some real-variable properties of these Hardy spaces. More precisely, the authors first establish the bounded joint H functional calculus in these Hardy spaces and prove that the abstract Riesz transform Dmi(L1+L2)-1/2 is bounded from HLip(n) to the classical Hardy space Hp(n) for all p(nn+mi,1], where i{1,2}. Moreover, for all p(0,1], the authors show that HL1+L˜2p(n)=HL1p(n)+HL2p(n) and give a sufficient condition to guarantee HL1p(n)HL2p(n).

Keywords: Hardy space; functional calculus; off-diagonal estimate; Riesz transform; higher order elliptic operator

MSC: 42B35; 42B30; 47A60; 35K08; 47B06; 35J48

About the article

Received: 2013-06-16

Published Online: 2013-12-05

Published in Print: 2015-09-01


Funding Source: Fundamental Research Funds for the Central Universities

Award identifier / Grant number: 2012YBXS16

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11271175

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11171027

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11361020

Funding Source: Specialized Research Fund for the Doctoral Program of Higher Education of China

Award identifier / Grant number: 20120003110003


Citation Information: Forum Mathematicum, Volume 27, Issue 5, Pages 2775–2824, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2013-0103.

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Jun Cao, Svitlana Mayboroda, and Dachun Yang
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