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Riesz transform characterizations of Musielak-Orlicz-Hardy spaces


Authors: Jun Cao, Der-Chen Chang, Dachun Yang and Sibei Yang
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
Published electronically: February 2, 2016
MathSciNet review: 3471083
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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

$\displaystyle 0<i(\varphi )\le I(\varphi )\le 1$

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$.

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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
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
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

DOI: https://doi.org/10.1090/tran/6556
Keywords: Riesz transform, harmonic function, Cauchy-Riemann equation, Musielak-Orlicz-Hardy space
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
Article copyright: © Copyright 2016 American Mathematical Society