Anisotropic Variable Hardy-Lorentz Spaces and Their Real Interpolation

Abstract

Let $p(\cdot ):{\mathbb{R}}^n\to (0,\infty )$ be a variable exponent function satisfying the globally log-Hölder 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 $H_A^{p(\cdot ),q}({\mathbb{R}}^n)$ 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 $H_A^{p(\cdot ),q}({\mathbb{R}}^n)$, respectively, in terms of the atom and the Lusin area function. As an application, the authors prove that the anisotropic variable Hardy-Lorentz space $H_A^{p(\cdot ),q}({\mathbb{R}}^n)$ severs as the intermediate space between the anisotropic variable Hardy space $H_A^{p(\cdot )}({\mathbb{R}}^n)$ and the space $L^{\infty }({\mathbb{R}}^n)$ via the real interpolation. This, together with a special case of the real interpolation theorem of H. Kempka and J. Vybíral on the variable Lorentz space, further implies the coincidence between $H_A^{p(\cdot ),q}({\mathbb{R}}^n)$ and the variable Lorentz space $L^{p(\cdot ),q}({\mathbb{R}}^n)$ when ${\mathrm{essinf}}_{x\in {\mathbb{R}}^n}p(x)\in (1,\infty )$.

42 pages, Submitted

References in corpus (1)

• Fractional, Maximal and Singular Operators in Variable Exponent Lorentz Spaces