Littlewood–Paley and Finite Atomic Characterizations of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Abstract

Let $$p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]$$ be a variable exponent function satisfying the globally log-Holder continuous condition, $$q\in (0,\infty ]$$ and A be a general expansive matrix on $${\mathbb {R}}^n$$ . Let $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ by means of the Littlewood–Paley g-function or the Littlewood–Paley $$g_\lambda ^*$$ -function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space $$L^{p(\cdot ),q}({\mathbb {R}}^n)$$ . Moreover, the finite atomic characterization of $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from $$H^{p(\cdot ),q}_A({\mathbb {R}}^n)$$ into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from $$H^{p(\cdot ),q}_A({\mathbb {R}}^n)$$ to $$L^{p(\cdot ),q}({\mathbb {R}}^n)$$ 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.