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Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces


Authors: Yiyu Liang and Dachun Yang
Journal: Trans. Amer. Math. Soc. 367 (2015), 3225-3256
MSC (2010): Primary 42B25; Secondary 42B30, 42B35, 46E30
DOI: https://doi.org/10.1090/S0002-9947-2014-06180-1
Published electronically: October 10, 2014
MathSciNet review: 3314807
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Abstract: Let $ \varphi : \mathbb{R}^n\times [0,\infty )\to [0,\infty )$ be such that $ \varphi (x,\cdot )$ is an Orlicz function and $ \varphi (\cdot ,t)$ is a Muckenhoupt $ A_\infty (\mathbb{R}^n)$ weight uniformly in $ t$. In this article, for any $ \alpha \in (0,1]$ and $ s\in \mathbb{Z}_+$, the authors establish the $ s$-order intrinsic square function characterizations of $ H^{\varphi }(\mathbb{R}^n)$ in terms of the intrinsic Lusin area function $ S_{\alpha ,s}$, the intrinsic $ g$-function $ g_{\alpha ,s}$ and the intrinsic $ g_{\lambda }^*$-function $ g^\ast _{\lambda , \alpha ,s}$ with the best known range $ \lambda \in (2+2(\alpha +s)/n,\infty )$, which are defined via $ \mathop {\mathrm {Lip}}_\alpha ({\mathbb{R}}^n)$ functions supporting in the unit ball. A $ \varphi $-Carleson measure characterization of the Musielak-Orlicz Campanato space $ {\mathcal L}_{\varphi ,1,s}({\mathbb{R}}^n)$ is also established via the intrinsic function. To obtain these characterizations, the authors first show that these $ s$-order intrinsic square functions are pointwise comparable with those similar-looking $ s$-order intrinsic square functions defined via $ \mathop {\mathrm {Lip}}_\alpha ({\mathbb{R}}^n)$ functions without compact supports, which when $ s=0$ was obtained by M. Wilson. All these characterizations of $ H^{\varphi }(\mathbb{R}^n)$, even when $ s=0$,

$\displaystyle \varphi (x,t):=w(x)t^p\ {\rm for\ all}\ t\in [0,\infty )\ {\rm and}\ x\in {\mathbb{R}}^n$

with $ p\in (n/(n+\alpha ), 1]$ and $ w\in A_{p(1+\alpha /n)}(\mathbb{R}^n)$, also essentially improve the known results.

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Additional Information

Yiyu Liang
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: yyliang@mail.bnu.edu.cn

Dachun Yang
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: dcyang@bnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2014-06180-1
Keywords: Musielak-Orlicz function, Hardy space, intrinsic square function, Carleson measure.
Received by editor(s): February 12, 2013
Published electronically: October 10, 2014
Additional Notes: The second (corresponding) author was supported by the National Natural Science Foundation of China (Grant No. 11171027) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003).
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.