Anisotropic variable Hardy–Lorentz spaces and their real interpolation

Abstract

Let p(⋅):Rn→(0,∞] be a variable exponent function satisfying the globally log-Hölder continuous condition, q∈(0,∞] and A be a general expansive matrix on Rn. In this article, the authors first introduce the anisotropic variable Hardy–Lorentz space HAp(⋅),q(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 HAp(⋅),q(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 HAp(⋅),q(Rn) serves as the intermediate space between the anisotropic variable Hardy space HAp(⋅)(Rn) and the space L∞(Rn) 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 HAp(⋅),q(Rn) and the variable Lorentz space Lp(⋅),q(Rn) when essinfx∈Rnp(x)∈(1,∞).