Variable Anisotropic Hardy Spaces and Their Applications

Let p(·) Rⁿ → (0,∞] be a variable exponent function satisfying the globally log-Hölder continuous condition and A a general expansive matrix on Rⁿ. In this article, the authors first introduce the variable anisotropic Hardy space HAp(·)(Rⁿ) associated with A, via the non-tangential grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of HAp(·)(Rⁿ), respectively, by means of atoms, finite atoms, the Lusin area function, the Littlewood-Paley g-function or gλ^-function. As applications, the authors first establish a criterion on the boundedness of sublinear operators from Hp(·)A(Rⁿ) into a quasi-Banach space. Then, applying this criterion, the authors show that the maximal operators of the Bochner-Riesz and the Weierstrass means are bounded from Hp(·)A(Rⁿ) to Lp(·)(Rⁿ) and, as consequences, some almost everywhere and norm convergences of these Bochner-Riesz and Weierstrass means are also obtained. These results on the Bochner-Riesz and the Weierstrass means are new even in the isotropic case.