Atomic characterizations of variable Hardy spaces on domains and their applications

Abstract

Let $\varOmega$ be a proper open subset of ${\mathbb {R}}^n$ and $p(\cdot ):\varOmega \rightarrow (0,\infty )$ a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the author introduces the variable Hardy space $H^{p(\cdot )}(\varOmega )$ on $\varOmega$ by the radial maximal function and then characterize the space $H^{p(\cdot )}(\varOmega )$ via grand maximal functions and atoms. Moreover, the author introduces the variable $\rm {BMO}$ space $\rm {BMO}^{p(\cdot )}(\varOmega )$ and the variable Hölder space $\varLambda ^{p(\cdot ),\,q,\,d}(\varOmega )$ on $\varOmega$. As applications of atomic characterizations of $H^{p(\cdot )}(\varOmega )$, the author shows that $\varLambda ^{p(\cdot ),\,q,\,d}(\varOmega )$ is the dual space of $H^{p(\cdot )}(\varOmega )$. In particular, when $\varOmega$ is a bounded Lipschitz domain in ${\mathbb {R}}^n$, the author further obtains $H^{p(\cdot )}(\varOmega )=H^{p(\cdot )}_{r}(\varOmega )$, $\rm {BMO}^{p(\cdot )}(\varOmega ) =\rm {BMO}^{p(\cdot )}_z(\varOmega )$ and $\varLambda ^{p(\cdot ),\,q,\,0}(\varOmega )=\varLambda ^{p(\cdot ),\,q,\,0}_z(\varOmega )$ with equivalent (quasi-)norm. Here the variable Hardy space $H^{p(\cdot )}_{r}(\varOmega )$ is defined via restricting arbitrary elements of $H^{p(\cdot )}({\mathbb {R}}^n)$ to $\varOmega$, $\rm {BMO}^{p(\cdot )}_z(\varOmega ):=\{f\in \rm {BMO}^{p(\cdot )}({\mathbb {R}}^n):\ {{\,\rm{supp}\,}} (f)\subset {\overline{\varOmega }}\}$ and $\varLambda ^{p(\cdot ),\,q,\,d}_z(\varOmega ): =\{f\in \varLambda ^{p(\cdot ),\,q,\,d}({\mathbb {R}}^n):\ {{\,\rm{supp}\,}} (f)\subset {\overline{\varOmega }}\}$, where $H^{p(\cdot )}({\mathbb {R}}^n)$, $\rm {BMO}^{p(\cdot )}({\mathbb {R}}^n)$ and $\varLambda ^{p(\cdot ),\,q,\,d}({\mathbb {R}}^n)$, respectively, denote the variable Hardy space, the variable $\rm {BMO}$ space and the variable Hölder space on ${\mathbb {R}}^n$, and ${\overline{\varOmega }}$ denotes the closure of $\varOmega$ in ${\mathbb {R}}^n$. The above results extend the main results in Miyachi (Studia Math 95:205–228, 1990) to the case of variable exponents.