Riesz transform characterizations of Musielak-Orlicz-Hardy spaces
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- by Jun Cao, Der-Chen Chang, Dachun Yang and Sibei Yang PDF
- Trans. Amer. Math. Soc. 368 (2016), 6979-7018 Request permission
Abstract:
Let $\varphi$ be a Musielak-Orlicz function satisfying that, for any $(x, t)\in \mathbb {R}^n\times (0, \infty )$, $\varphi (\cdot , t)$ belongs to the Muckenhoupt weight class $A_\infty (\mathbb {R}^n)$ with the critical weight exponent $q(\varphi )\in [1, \infty )$ and $\varphi (x, \cdot )$ is an Orlicz function with \[ 0<i(\varphi )\le I(\varphi )\le 1\] which are, respectively, its critical lower type and upper type. In this article, the authors establish the Riesz transform characterizations of the Musielak-Orlicz-Hardy spaces $H_\varphi (\mathbb {R}^n)$ which are generalizations of weighted Hardy spaces and Orlicz-Hardy spaces. Precisely, the authors characterize $H_\varphi (\mathbb {R}^n)$ via all the first order Riesz transforms when $\frac {i(\varphi )}{q(\varphi )}>\frac {n-1}{n}$, and via all the Riesz transforms with the order not more than $m\in \mathbb {N}$ when $\frac {i(\varphi )}{q(\varphi )}>\frac {n-1}{n+m-1}$. Moreover, the authors also establish the Riesz transform characterizations of $H_\varphi (\mathbb {R}^n)$, respectively, by means of the higher order Riesz transforms defined via the homogenous harmonic polynomials or the odd order Riesz transforms. Even if when $\varphi (x,t):=tw(x)$ for all $x\in {\mathbb R}^n$ and $t\in [0,\infty )$, these results also widen the range of weights in the known Riesz characterization of the classical weighted Hardy space $H^1_w({\mathbb R}^n)$ obtained by R. L. Wheeden from $w\in A_1({\mathbb R}^n)$ into $w\in A_\infty ({\mathbb R}^n)$ with the sharp range $q(w)\in [1,\frac n{n-1})$, where $q(w)$ denotes the critical index of the weight $w$.References
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Additional Information
- Jun Cao
- Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
- Address at time of publication: Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, People’s Republic of China
- Email: caojun1860@zjut.edu.cn
- Der-Chen Chang
- Affiliation: Department of Mathematics and Department of Computer Science, Georgetown University, Washington DC 20057 – and – Department of Mathematics, Fu Jen Catholic University, Taipei 242, Taiwan
- MR Author ID: 47325
- Email: chang@georgetown.edu
- 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
- Sibei Yang
- Affiliation: School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
- Email: yangsb@lzu.edu.cn
- Received by editor(s): January 27, 2014
- Received by editor(s) in revised form: August 27, 2014
- Published electronically: February 2, 2016
- Additional Notes: The third author is the corresponding author
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6979-7018
- MSC (2010): Primary 47B06; Secondary 42B20, 42B30, 42B35, 46E30
- DOI: https://doi.org/10.1090/tran/6556
- MathSciNet review: 3471083