We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
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.
