Anisotropic Hardy-Lorentz spaces and their applications

By Jun Liu, Dachun Yang, Wen Yuan. 2015

74 citations recorded by the source corpus.

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 $$H_A^{{p_1},{q_1}}\left( {{\mathbb{R}^n}} \right)$$ 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 (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 $$\delta \in \left( {0,\frac{{In{\lambda _ - }}}{{Inb}}} \right]\;,\;p \in \left( {\frac{1}{{1 + \delta }},\;1} \right]\;and\;q \in \left( {0,\;\infty } \right]$$ and q ∈ (0,∞], as well as the boundedness of some Calderón-Zygmund operators from HAp,q (ℝn) to Lp,∞(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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0c7c0b3730402eac4650d07534fd1423b94be24c

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In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. Yangyang Zhang
4. Jun Liu
5. Songbai Wang
6. Long Huang
7. Ferenc Weisz
8. Yoshihiro Sawano

In Mathematical Physics

1. Dachun Yang
2. Wen Yuan
3. Jun Liu
4. Yangyang Zhang
5. Songbai Wang
6. Long Huang
7. Ferenc Weisz
8. Yoshihiro Sawano

Citations in the corpus

• Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications (2018, methodology): “For more progresses about the theory of anisotropic function spaces on R, we refer the reader to [11, 25, 45, 48, 49, 51, 53, 54, 55] for anisotropic Hardy-type spaces and to [7, 8, 9, 46, 47, 56] for anisotropic Besov and TriebelLizorkin spaces.”
• Anisotropic Variable Hardy-Lorentz Spaces and Their Real Interpolation (2017, background): “To this end, we first recall the notion of the anisotropic Lusin area function (see [48, 50]).”
• Real-variable characterizations of Orlicz-slice Hardy spaces (2018, background): “In recent decades, in order to meet the requirements arising in the study of the boundedness of operators, partial differential equations and some other fields, various variants of Hardy spaces have been introduced and developed, such as weak Hardy spaces (see, for example, [17, 20]), Hardy-Lorentz ”
• Variable Anisotropic Hardy Spaces and Their Applications (2017, methodology): “We begin with recalling the following notion of Littlewood-Paley functions (see, for example, [40,45]).”