Discrete para-product operators on variable Hardy spaces

Abstract

Let $p(\cdot ):{\mathbb{R}}^n\to (0,\infty )$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of para-product operators ${\pi }_b$ on variable Hardy spaces $H^{p(\cdot )}({\mathbb{R}}^n)$, where $b\in BMO({\mathbb{R}}^n)$. 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 $\frac{n}{n+ε}<\mbox{essinf}_{x\in\mathbb R^n} p\le \mbox{esssup}_{x\in\mathbb R^n} p\le 1$, $\epsilon$ 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 and almost orthogonal estimates. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

17 pages