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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces
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by Yiyu Liang and Dachun Yang PDF
Trans. Amer. Math. Soc. 367 (2015), 3225-3256 Request permission

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 $\mathrm {Lip}_\alpha ({\mathbb R}^n)$ 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 $\mathrm {Lip}_\alpha ({\mathbb R}^n)$ functions without compact supports, which when $s=0$ was obtained by M. Wilson. All these characterizations of $H^{\varphi }(\mathbb R^n)$, even when $s=0$, \[ \varphi (x,t):=w(x)t^p\ \textrm {for\ all}\ t\in [0,\infty )\ \textrm {and}\ x\in {\mathbb R}^n\] 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
  • MR Author ID: 946733
  • 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
  • MR Author ID: 317762
  • Email: dcyang@bnu.edu.cn
  • 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).
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 3314807