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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 27, Issue 5

# 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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• Other articles by this author:
/ 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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• Other articles by this author:
/ 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 ${L}_{1},{L}_{2}$ be a pair of one-to-one commuting sectorial operators such that each Li for $i\in \left\{1,2\right\}$ satisfies the mi order L2 off-diagonal estimates and ${m}_{1}\ge {m}_{2}>0$. Let ${H}_{{L}_{i}}^{p}\left({ℝ}^{n}\right)$, $i\in \left\{1,2\right\}$, and ${H}_{{L}_{1}+{\stackrel{˜}{L}}_{2}}^{p}\left({ℝ}^{n}\right)$ be the Hardy spaces associated, respectively, to the operators Li and ${L}_{1}+{\stackrel{˜}{L}}_{2}$, where ${\stackrel{˜}{L}}_{2}:={L}_{2}^{{m}_{1}/{m}_{2}}$ 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 ${D}^{{m}_{i}}{\left({L}_{1}+{L}_{2}\right)}^{-1/2}$ is bounded from ${H}_{{L}_{i}}^{p}\left({ℝ}^{n}\right)$ to the classical Hardy space ${H}^{p}\left({ℝ}^{n}\right)$ for all $p\in \left(\frac{n}{n+{m}_{i}},1\right]$, where $i\in \left\{1,2\right\}$. Moreover, for all $p\in \left(0,1\right]$, the authors show that ${H}_{{L}_{1}+{\stackrel{˜}{L}}_{2}}^{p}\left({ℝ}^{n}\right)={H}_{{L}_{1}}^{p}\left({ℝ}^{n}\right)+{H}_{{L}_{2}}^{p}\left({ℝ}^{n}\right)$ and give a sufficient condition to guarantee ${H}_{{L}_{1}}^{p}\left({ℝ}^{n}\right)\subset {H}_{{L}_{2}}^{p}\left({ℝ}^{n}\right)$.

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

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

Award identifier / Grant number: 20120003110003

Citation Information: Forum Mathematicum, Volume 27, Issue 5, Pages 2775–2824, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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