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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 28, Issue 5

# Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators

Jun Cao
• School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China. Current address: Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023
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• Other articles by this author:
/ Svitlana Mayboroda
/ Dachun Yang
• Corresponding author
• School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China
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• Other articles by this author:
Published Online: 2015-10-01 | DOI: https://doi.org/10.1515/forum-2014-0127

## Abstract

Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and $\left({p}_{-}\left(L\right),{p}_{+}\left(L\right)\right)$ be the maximal interval of exponents $q\in \left[1,\mathrm{\infty }\right]$ such that the semigroup ${\left\{{e}^{-tL}\right\}}_{t>0}$ is bounded on ${L}^{q}\left({ℝ}^{n}\right)$. In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces ${H}_{L}^{p}\left({ℝ}^{n}\right)$ for all $p\in \left(0,{p}_{+}\left(L\right)\right)$, which when $p=1$, answers a question asked by Deng, Ding and Yao in [21]. Moreover, the authors characterize ${H}_{L}^{p}\left({ℝ}^{n}\right)$ via various versions of square functions and Lusin-area functions associated to the operator L.

MSC 2010: 42B30; 42B20; 42B35; 42B37; 46E30; 35J48; 47B06; 47B38

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Revised: 2015-04-21

Published Online: 2015-10-01

Published in Print: 2016-09-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS 1220089

Award identifier / Grant number: DMS 1344235

Award identifier / Grant number: DMR 0212302

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11501506

Award identifier / Grant number: 11571039

Award identifier / Grant number: 11361020

J. Cao is supported by the National Natural Science Foundation of China (grant no. 11501506) and the Natural Science Foundation of Zhejiang University of Technology (grant no. 2014XZ011). S. Mayboroda was partially supported by the NSF grants DMS 1220089 (CAREER), DMS 1344235 (INSPIRE), DMR 0212302 (UMN MRSEC Seed grant) and the Alfred P. Sloan Fellowship. D. Yang is supported by the National Natural Science Foundation of China (grant no. 11571039 and 11361020), the Specialized Research Fund for the Doctoral Program of Higher Education of China (grant no. 20120003110003) and the Fundamental Research Funds for the Central Universities of China (grant no. 2013YB60 and 2014KJJCA10).

Citation Information: Forum Mathematicum, Volume 28, Issue 5, Pages 823–856, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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