Variable weak Hardy spaces and their applications

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• 作者：Xianjie Yan ^{xianjieyan@mail.bnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Dachun Yang ; ^{dcyang@bnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Wen Yuan ^{wenyuan@bnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Ciqiang Zhuo ^{cqzhuo@mail.bnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 42B30 ; secondary ; 42B25 ; 42B20 ; 42B35 ; 46E30
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 November 2016
• 年：2016
• 卷：271
• 期：10
• 页码：2822-2887
• 全文大小：932 K

文摘

Let $p(\cdot ):{\mathbb{R}}^n\to (0,\infty )$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first introduce the variable weak Hardy space on Rⁿ${\mathbb{R}}^n$, WH^p(⋅)(Rⁿ)${\mathit{WH}}^{p(\cdot )}({\mathbb{R}}^n)$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of WH^p(⋅)(Rⁿ)${\mathit{WH}}^{p(\cdot )}({\mathbb{R}}^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood–Paley g  -function or $g_{\lambda }^{⁎}$-function. As an application, the authors establish the boundedness of convolutional δ-type and non-convolutional γ  -order Calderón–Zygmund operators from H^p(⋅)(Rⁿ)$H^{p(\cdot )}({\mathbb{R}}^n)$ to WH^p(⋅)(Rⁿ)${\mathit{WH}}^{p(\cdot )}({\mathbb{R}}^n)$ including the critical case when p₋=n/(n+δ)$p_{-}=n/(n+\delta )$ or when p₋=n/(n+γ)$p_{-}=n/(n+\gamma )$, where $p_{-}:={\mathrm{ess}\inf }_{x\in {\mathbb{R}}^n}p(x)$.