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.
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Liu, J., Yang, D. & Yuan, W. Anisotropic Hardy-Lorentz spaces and their applications. Sci. China Math. 59, 1669–1720 (2016). https://doi.org/10.1007/s11425-016-5157-y
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DOI: https://doi.org/10.1007/s11425-016-5157-y
Keywords
- Lorentz space
- anisotropic Hardy-Lorentz space
- expansive matrix
- Calderón reproducing formula
- grand maximal function
- atom
- molecule
- Calderón-Zygmund operator