Bulletin of the
Korean Mathematical Society BKMS

Let $\Omega$ be a proper open subset of $\rn$ and $p(\cdot):\Omega\rightarrow(0,\,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition. In this article, the author introduces the ``geometrical" variable Hardy spaces $H_r^{p(\cdot)}(\Omega)$ and $H_z^{p(\cdot)}(\Omega)$ on $\Omega$, and then obtains the grand maximal function characterizations of $H_r^{p(\cdot)}(\Omega)$ and $H_z^{p(\cdot)}(\Omega)$ when $\Omega$ is a strongly Lipschitz domain of $\rn$. Moreover, the author further introduces the ``geometrical" variable local Hardy spaces $h_r^{p(\cdot)}(\Omega)$, and then establishes the atomic characterization of $h_r^{p(\cdot)}(\Omega)$ when $\Omega$ is a bounded Lipschitz domain of $\rn$.

Keywords: Variable Hardy space, grand maximal function, atom, Lipschitz domains

MSC numbers: Primary 42B30; Secondary 42B25, 46A20, 42B35, 46E30

Supported by: This work is supported by the National Natural Science Foundation of China (Grant No. 11871254)