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Discrete Paraproduct Operators on Variable Hardy Spaces
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 2 / June 2020
- Published online by Cambridge University Press:
- 04 October 2019, pp. 304-317
- Print publication:
- June 2020
-
- Article
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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.
