Anisotropic Hardy-Lorentz spaces and their applications

Let p ∈ (0, 1], q ∈ (0,∞] and A be a general expansive matrix on ℝn. We introduce the anisotropic Hardy-Lorentz space HAp,q (ℝn) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on ℝn. As applications, we first prove that HAp,q (ℝn) is an intermediate space between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_A^{{p_1},{q_1}}\left( {{\mathbb{R}^n}} \right)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_A^{{p_2},{q_2}}\left( {{\mathbb{R}^n}} \right)$$\end{document} with 0 < p1 < p < p2 < ∞ and q1, q, q2 ∈ (0,∞], and also between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_A^{p,{q_1}}\left( {{\mathbb{R}^n}} \right)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_A^{p,{q_2}}\left( {{\mathbb{R}^n}} \right)$$\end{document} with p ∈ (0,∞) and 0 < q1 < q < q2 ⩽ ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from HAp,q (ℝn) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calderón-Zygmund operators from HAp (ℝn) to the weak Lebesgue space Lp,∞(ℝn) (or to HAp,∞ (ℝn) in the critical case, from HAp (ℝn) to Lp,q(ℝn) (or to HAp (ℝn)) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \in \left( {0,\frac{{In{\lambda _ - }}}{{Inb}}} \right]\;,\;p \in \left( {\frac{1}{{1 + \delta }},\;1} \right]\;and\;q \in \left( {0,\;\infty } \right]$$\end{document} and q ∈ (0,∞], as well as the boundedness of some Calderón-Zygmund operators from HAp,q (ℝn) to Lp,∞(ℝn), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\;: \in \left| {\det \;A} \right|$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda _ - }\;: = \;\min \left\{ {\left| \lambda \right|\;:\lambda \; \in \;\sigma \left( A \right)} \right\}$$\end{document} and σ(A) denotes the set of all eigenvalues of A.