| Abstract: | Let p ∈ (0, 1], q ∈ (0,∞] and A be a general expansive matrix on ?n. We introduce the anisotropic Hardy-Lorentz space HAp,q (?n) 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 ?n. As applications, we first prove that HAp,q (?n) is an intermediate space between \(H_A^{{p_1},{q_1}}\left( {{\mathbb{R}^n}} \right)\) and \(H_A^{{p_2},{q_2}}\left( {{\mathbb{R}^n}} \right)\) with 0 < p1 < p < p2 < ∞ and q1, q, q2 ∈ (0,∞], and also between \(H_A^{p,{q_1}}\left( {{\mathbb{R}^n}} \right)\) and \(H_A^{p,{q_2}}\left( {{\mathbb{R}^n}} \right)\) 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 (?n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calderón-Zygmund operators from HAp (?n) to the weak Lebesgue space Lp,∞(?n) (or to HAp,∞ (?n) in the critical case, from HAp (?n) to Lp,q(?n) (or to HAp (?n)) with \(\delta \in \left( {0,\frac{{In{\lambda _ - }}}{{Inb}}} \right],p \in \left( {\frac{1}{{1 + \delta }},1} \right]andq \in \left( {0,\infty } \right]\) and q ∈ (0,∞], as well as the boundedness of some Calderón-Zygmund operators from HAp,q (?n) to Lp,∞(?n), where \(b: \in \left| {\det A} \right|\), \({\lambda _ - }: = \min \left\{ {\left| \lambda \right|:\lambda \in \sigma \left( A \right)} \right\}\) and σ(A) denotes the set of all eigenvalues of A. |
