Anisotropic Hardy-Lorentz spaces and their applications

Abstract

Let p ∈ (0, 1], q ∈ (0,∞] and A be a general expansive matrix on ℝⁿ. We introduce the anisotropic Hardy-Lorentz space H ^p,q _A (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 H ^p,q _A (ℝⁿ) 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 < p ₁ < p < p ₂ < ∞ and q ₁, q, q ₂ ∈ (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 < q ₁ < q < q ₂ ⩽ ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H ^p,q _A (ℝⁿ) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calderón-Zygmund operators from H ^p _A (ℝⁿ) to the weak Lebesgue space L ^p,∞(ℝⁿ) (or to H ^p,∞ _A (ℝⁿ) in the critical case, from H ^p _A (ℝⁿ) to L ^p,q(ℝⁿ) (or to H ^p _A (ℝⁿ)) with $\delta \in \left( {0,\frac{{In{\lambda _ - }}}{{Inb}}} \right]\;,\;p \in \left( {\frac{1}{{1 + \delta }},\;1} \right]\;and\;q \in \left( {0,\;\infty } \right]$ and q ∈ (0,∞], as well as the boundedness of some Calderón-Zygmund operators from H ^p,q _A (ℝⁿ) to L ^p,∞(ℝⁿ), 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.