Anisotropic Hardy-Lorentz spaces and their applications  

作　　者：LIU Jun YANG DaChun YUAN Wen 

机构地区：[1]School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education

出　　处：《Science China Mathematics》2016年第9期1669-1720,共52页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 11571039, 11361020 and 11471042)

摘　　要：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(R^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 Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H^(p1,q1)_A(Rn) and H^(p2,q2)_A(R^n) with 0 < p1 < p < p2 < ∞ and q1, q, q2 ∈(0, ∞], and also between H^(p,q1)_A(Rn) and H^(p,q2)_A(R^n) 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 H^(p,q)_A(R^n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H^(p,∞)_A(R^n) to the weak Lebesgue space L^(p,∞)(R^n)(or to H^p_A(R^n)) in the ln λcritical case, from H^(p,q)_A(R^n) to L^(p,q)(R^n)(or to H^(p,q)_A(R^n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H^(p,q)_A(R^n) to L^(p,∞)(R^n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.Let p ∈（0, 1], q ∈（0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H^p,qA（R^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 Rn.As applications, we first prove that Hp,q A（Rn） is an intermediate space between H^（p1,q1）_A（Rn） and H^（p2,q2）_A（R^n） with 0 〈 p1 〈 p 〈 p2 〈 ∞ and q1, q, q2 ∈（0, ∞], and also between H^（p,q1）_A（Rn） and H^（p,q2）_A（R^n） 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 H^p,qA（R^n） into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H^（p,∞）_A（R^n） to the weak Lebesgue space L^（p,∞）（R^n）（or to H^p_A（R^n）） in the ln λcritical case, from H^p,qA（R^n） to L^（p,q）（R^n）（or to H^p,qA（R^n）） with δ∈（0,（lnλ）/（ln b）], p ∈（1/（1＋,δ）,1] and q ∈（0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H^p,qA（R^n） to L^（p,∞）（R^n）, where b ：= ｜ det A｜,λ_：= min{｜λ｜ ： λ∈σ（A）} and σ（A） denotes the set of all eigenvalues of A.

关 键 词：Lorentz space anisotropic Hardy-Lorentz space expansive matrix Calder′on reproducing formula grand maximal function atom molecule Calder′on-Zygmund operator 

分 类 号：O177[理学—数学]