Anisotropic Hardy-Lorentz Spaces and Their Applications

Abstract

Let $p\in (0,1]$, $q\in (0,\infty ]$ and $A$ be a general expansive matrix on ${\mathbb{R}}^n$. The authors introduce the anisotropic Hardy-Lorentz space $H_A^{p,q}({\mathbb{R}}^n)$ associated with $A$ via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic or the molecular decompositions, the radial or the non-tangential maximal functions, or the finite atomic decompositions. All these characterizations except the $\infty$-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on ${\mathbb{R}}^n$. As applications, the authors first prove that $H_A^{p,q}({\mathbb{R}}^n)$ is an intermediate space between $H_A^{p_1,q_1}({\mathbb{R}}^n)$ and $H_A^{p_2,q_2}({\mathbb{R}}^n)$ with $0<p_1<p<p_2<\infty$ and $q_1,q,q_2\in (0,\infty ]$, and also between $H_A^{p,q_1}({\mathbb{R}}^n)$ and $H_A^{p,q_2}({\mathbb{R}}^n)$ with $p\in (0,\infty )$ and $0<q_1<q<q_2\le \infty$ in the real method of interpolation. The authors then establish a criterion on the boundedness of sublinear operators from $H_A^{p,q}({\mathbb{R}}^n)$ into a quasi-Banach space; moreover, the authors obtain the boundedness of $\delta$-type Calderón-Zygmund operators from $H_A^p({\mathbb{R}}^n)$ to the weak Lebesgue space $L^{p,\infty }({\mathbb{R}}^n)$ (or $H_A^{p,\infty }({\mathbb{R}}^n)$) in the critical case, from $H_A^{p,q}({\mathbb{R}}^n)$ to $L^{p,q}({\mathbb{R}}^n)$ (or $H_A^{p,q}({\mathbb{R}}^n)$) with $\delta \in (0,\frac{\ln {\lambda }_{-}}{\ln b}]$, $p\in (\frac{1}{1+\delta },1]$ and $q\in (0,\infty ]$, as well as the boundedness of some Calderón-Zygmund operators from $H_A^{p,q}({\mathbb{R}}^n)$ to $L^{p,\infty }({\mathbb{R}}^n)$, where $b:=|\det A|$, ${\lambda }_{-}:=\min \{|\lambda |:\lambda \in \sigma (A)\}$ and $\sigma (A)$ denotes the set of all eigenvalues of $A$.

68 pages; submitted

References in corpus (4)

• Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces
• Hardy spaces $H^p$ over non-homogeneous metric measure spaces and their applications
• Littlewood-Paley Characterizations of Anisotropic Hardy-Lorentz Spaces
• Sharp decay estimates in Lorentz spaces for nonnegative Schrödinger heat semigroups

Cited by in corpus (5)

• Atomic and Littlewood-Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications
• Anisotropic Variable Hardy-Lorentz Spaces and Their Real Interpolation
• Real-Variable Characterizations of New Anisotropic Mixed-Norm Hardy Spaces
• Molecular Decomposition of Anisotropic Hardy Spaces with Variable Exponents
• Real-Variable Characterizations of Orlicz-Slice Hardy Spaces