Hardy spaces H^p over non-homogeneous metric measure spaces and their applications  

作　　者：FU Xing LIN Hai Bo YANG Da Chun YANG Dong Yong 

机构地区：[1]School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems,Ministry of Education [2]College of Science, China Agricultural University [3]School of Mathematical Sciences, Xiamen University

出　　处：《Science China Mathematics》2015年第2期309-388,共80页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11301534,11171027,11361020 and 11101339);Da Bei Nong Education Fund(Grant No.1101-2413002);the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120003110003);the Fundamental Research Funds for Central Universities of China(Grant Nos.2012LYB26,2012CXQT09,2013YB60 and 2014KJJCA10)

摘　　要：Let(X,d,)be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions.Let ρ∈(1,∞),0<p≤1≤q≤∞,p≠q,γ∈[1,∞)and ∈∈(0,∞).In this paper,the authors introduce the atomic Hardy space Hp,q,γ atb,ρ(μ)and the molecular Hardy space Hp,q,γ,mb,ρ∈(μ)via the discrete coefficient K(ρ),p B,S,and prove that the Calder′on-Zygmund operator is bounded from Hp,q,γ,δmb,ρ(μ)(or Hp,q,γatb,ρ(μ))into Lp(μ),and from Hp,q,γ+1atb,ρ(ρ+1)(μ)into H p,q,γ,12(δ-νp+ν)mb,ρ(μ).The boundedness of the generalized fractional integral Tβ(β∈(0,1))from Hp1,q,γ,θmb,ρ(μ)(or Hp1,q,γatb,ρ(μ))into Lp2(μ)with 1/p2=1/p1-β is also established.The authors also introduce theρ-weakly doubling condition,withρ∈(1,∞),of the measure and construct a non-doubling measure satisfying this condition.If isρ-weakly doubling,the authors further introduce the Campanato space Eα,qρ,η,γ(μ)and show that Eα,qρ,η,γ(μ)is independent of the choices ofρ,η,γand q;the authors then introduce the atomic Hardy space Hp,q,γatb,ρ(μ)and the molecular Hardy space Hp,q,γ,mb,ρ(μ),which coincide with each other;the authors finally prove that Hp,q,γatb,ρ(μ)is the predual of E1/p-1,1ρ,ρ,1(μ).Moreover,if is doubling,the authors show that Eα,qρ,η,γ(μ)and the Lipschitz space Lipα,q(μ)(q∈[1,∞)),or Hp,q,γatb,ρ(μ)and the atomic Hardy space Hp,q at(μ)(q∈(1,∞])of Coifman and Weiss coincide.Finally,if(X,d,)is an RD-space(reverse doubling space)with(X)=∞,the authors prove that Hp,q,γatb,ρ(μ),Hp,q,γ,mb,ρ(μ)and Hp,q at(μ)coincide for any q∈(1,2].In particular,when(X,d,):=(RD,||,dx)with dx being the D-dimensional Lebesgue measure,the authors show that spaces Hp,q,γatb,ρ(μ),Hp,q,γ,mb,ρ(μ),Hp,q,γatb,ρ(μ)and Hp,q,γ,mb,ρ(μ)all coincide with Hp(RD)for any q∈(1,∞).Let（X,d,）be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions.Let ρ∈（1,∞）,0〈p≤1≤q≤∞,p≠q,γ∈[1,∞）and ∈ ∈（0,∞）.In this paper,the authors introduce the atomic Hardy space Hp,q,γ atb,ρ（μ）and the molecular Hardy space Hp,q,γ,mb,ρ ∈（μ）via the discrete coefficient K（ρ）,p B,S,and prove that the Calder′on-Zygmund operator is bounded from Hp,q,γ,δmb,ρ（μ）（or Hp,q,γatb,ρ（μ））into Lp（μ）,and from Hp,q,γ＋1atb,ρ（ρ＋1）（μ）into H p,q,γ,12（δ-νp＋ν）mb,ρ（μ）.The boundedness of the generalized fractional integral Tβ（β∈（0,1））from Hp1,q,γ,θmb,ρ（μ）（or Hp1,q,γatb,ρ（μ））into Lp2（μ）with 1/p2=1/p1-β is also established.The authors also introduce theρ-weakly doubling condition,withρ∈（1,∞）,of the measure and construct a non-doubling measure satisfying this condition.If isρ-weakly doubling,the authors further introduce the Campanato space Eα,qρ,η,γ（μ）and show that Eα,qρ,η,γ（μ）is independent of the choices ofρ,η,γand q;the authors then introduce the atomic Hardy space Hp,q,γatb,ρ（μ）and the molecular Hardy space Hp,q,γ,mb,ρ（μ）,which coincide with each other;the authors finally prove that Hp,q,γatb,ρ（μ）is the predual of E1/p-1,1ρ,ρ,1（μ）.Moreover,if is doubling,the authors show that Eα,qρ,η,γ（μ）and the Lipschitz space Lipα,q（μ）（q∈[1,∞））,or Hp,q,γatb,ρ（μ）and the atomic Hardy space Hp,q at（μ）（q∈（1,∞]）of Coifman and Weiss coincide.Finally,if（X,d,）is an RD-space（reverse doubling space）with（X）=∞,the authors prove that Hp,q,γatb,ρ（μ）,Hp,q,γ,mb,ρ（μ）and Hp,q at（μ）coincide for any q∈（1,2].In particular,when（X,d,）：=（RD,｜｜,dx）with dx being the D-dimensional Lebesgue measure,the authors show that spaces Hp,q,γatb,ρ（μ）,Hp,q,γ,mb,ρ（μ）,Hp,q,γatb,ρ（μ）and Hp,q,γ,mb,ρ（μ）all coincide wit

关 键 词：non-homogeneous metric measure space ρ-weakly doubling measure Hardy space Campanato space Lipschitz space Calder′on-Zygmund operator atomic block molecular block 

分 类 号：O177[理学—数学]