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-Holder 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 Holder 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 Holder 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.