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DISCRETE PARA-PRODUCT OPERATORS ON VARIABLE HARDY SPACES
Canadian Mathematical Bulletin ( IF 0.5 ) Pub Date : 2019-10-04 , DOI: 10.4153/s0008439519000298
Jian Tan

Let $p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )$ be a variable exponent function satisfying the globally log-Holder 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 Calderon–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 Calderon’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.

中文翻译:

可变 Hardy 空间上的离散 Para-product 算子

令 $p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )$ 为满足全局对数保持器连续条件的变指数函数。在本文中,我们获得了可变哈代空间 $H^{p(\cdot )}(\mathbb{R}^{n}) 上的副乘积算子 $\unicode[STIX]{x1D70B}_{b}$ 的有界性$ ,其中 $b\in \text{BMO}(\mathbb{R}^{n})$ 。作为一个应用,我们证明了非卷积类型的 Calderon-Zygmund 算子 $T$ 在 $H^{p(\cdot )}(\mathbb{R}^{n})$ 上有界当且仅当 $T^ {\ast }1=0$ ,其中 $\frac{n}{n+\unicode[STIX]{x1D716}}<\text{ess inf}_{x\in \mathbb{R}^{n}}p \leqslant \text{ess sup}_{x\in \mathbb{R}^{n}}p\leqslant 1$ 和 $\unicode[STIX]{x1D716}$ 是 $T$ 内核的常规指数。我们的方法依赖于 Calderon 复制公式的离散版本,离散的 Littlewood-Paley-Stein 理论,几乎正交的估计和可变指数分析技术。通过使用我们的方法,这些结果仍然适用于同构空间上的可变哈代空间。
更新日期:2019-10-04
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