Variable Weak Hardy Spaces and Their Applications

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 $\mathbb R^n$, $W\!H^{p(\cdot)}(\mathbb R^n)$, 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 $W\!H^{p(\cdot)}(\mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_λ^\ast$-function. As an application, the authors establish the boundedness of convolutional $δ$-type and non-convolutional $γ$-order Calderón-Zygmund operators from $H^{p(\cdot)}(\mathbb R^n)$ to $W\!H^{p(\cdot)}(\mathbb R^n)$ including the critical case $p_-={n}/{(n+δ)}$, where $p_-:=\mathop\mathrm{ess\,inf}_{x\in \rn}p(x).$

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