Anisotropic Variable Hardy-Lorentz Spaces and Their Real Interpolation (1705.05188v1)
Abstract: Let $p(\cdot):\ \mathbb Rn\to(0,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}n$. In this article, the authors first introduce the anisotropic variable Hardy-Lorentz space $H_A{p(\cdot),q}(\mathbb Rn)$ associated with $A$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain characterizations of $H_A{p(\cdot),q}(\mathbb Rn)$, respectively, in terms of the atom and the Lusin area function. As an application, the authors prove that the anisotropic variable Hardy-Lorentz space $H_A{p(\cdot),q}(\mathbb Rn)$ severs as the intermediate space between the anisotropic variable Hardy space $H_A{p(\cdot)}(\mathbb Rn)$ and the space $L\infty(\mathbb Rn)$ via the real interpolation. This, together with a special case of the real interpolation theorem of H. Kempka and J. Vyb\'iral on the variable Lorentz space, further implies the coincidence between $H_A{p(\cdot),q}(\mathbb Rn)$ and the variable Lorentz space $L{p(\cdot),q}(\mathbb Rn)$ when $\mathop\mathrm{essinf}_{x\in\mathbb{R}n}p(x)\in (1,\infty)$.