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Discrete Paraproduct Operators on Variable Hardy Spaces

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  04 October 2019

Jian Tan
Jian Tan*
Affiliation:
School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China Email: tanjian89@126.com
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Abstract

Let $p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of paraproduct operators $\unicode[STIX]{x1D70B}_{b}$ on variable Hardy spaces $H^{p(\cdot )}(\mathbb{R}^{n})$, where $b\in \text{BMO}(\mathbb{R}^{n})$. As an application, we show that non-convolution type Calderón–Zygmund operators $T$ are bounded on $H^{p(\cdot )}(\mathbb{R}^{n})$ if and only if $T^{\ast }1=0$, where $\frac{n}{n+\unicode[STIX]{x1D716}}<\text{ess inf}_{x\in \mathbb{R}^{n}}p\leqslant \text{ess sup}_{x\in \mathbb{R}^{n}}p\leqslant 1$ and $\unicode[STIX]{x1D716}$ is the regular exponent of kernel of $T$. Our approach relies on the discrete version of Calderón’s reproducing formula, discrete Littlewood–Paley–Stein theory, almost orthogonal estimates, and variable exponents analysis techniques. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

Keywords

variable Hardy spacesingular integralparaproduct operatordiscrete Littlewood–Paley–Stein theory

MSC classification

Primary: 42B30: $H^p$-spaces
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 2 , June 2020 , pp. 304 - 317
Copyright
© Canadian Mathematical Society 2019

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Footnotes

The project was sponsored by the Natural Science Foundation of Jiangsu Province of China (grant no. BK20180734), National Natural Science Foundation of China (No.11901309), the Natural Science Research of Jiangsu Higher Education Institutions of China (grant no. 18KJB110022), and Nanjing University of Posts and Telecommunications Science Foundation (grant nos. NY219114, NY217151).

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