Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces  

作　　者：LIN HaiBo YANG DaChun 

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

出　　处：《Science China Mathematics》2014年第1期123-144,共22页中国科学：数学（英文版）

基　　金：supported by the Mathematical Tianyuan Youth Fund of the National Natural Science Foundation of China (Grant No. 11026120);Chinese Universities Scientific Fund (Grant No. 2011JS043);National Natural Science Foundation of China (Grant Nos. 11171027 and 11361020);the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003)

摘　　要：Let(X,d,μ)be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of Hyto¨nen.We prove that the L p(μ)-boundedness with p∈(1,∞)of the Marcinkiewicz integral is equivalent to either of its boundedness from L1(μ)into L1,∞(μ)or from the atomic Hardy space H1(μ)into L1(μ).Moreover,we show that,if the Marcinkiewicz integral is bounded from H1(μ)into L1(μ),then it is also bounded from L∞(μ)into the space RBLO(μ)(the regularized BLO),which is a proper subset of RBMO(μ)(the regularized BMO)and,conversely,if the Marcinkiewicz integral is bounded from L∞b(μ)(the set of all L∞(μ)functions with bounded support)into the space RBMO(μ),then it is also bounded from the finite atomic Hardy space H1,∞fin(μ)into L1(μ).These results essentially improve the known results even for non-doubling measures.Let（X,d,μ）be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of Hyto¨nen.We prove that the L p（μ）-boundedness with p∈（1,∞）of the Marcinkiewicz integral is equivalent to either of its boundedness from L1（μ）into L1,∞（μ）or from the atomic Hardy space H1（μ）into L1（μ）.Moreover,we show that,if the Marcinkiewicz integral is bounded from H1（μ）into L1（μ）,then it is also bounded from L∞（μ）into the space RBLO（μ）（the regularized BLO）,which is a proper subset of RBMO（μ）（the regularized BMO）and,conversely,if the Marcinkiewicz integral is bounded from L∞b（μ）（the set of all L∞（μ）functions with bounded support）into the space RBMO（μ）,then it is also bounded from the finite atomic Hardy space H1,∞fin（μ）into L1（μ）.These results essentially improve the known results even for non-doubling measures.

关 键 词：UPPER doubling geometrically doubling Marcinkiewicz integral atomic HARDY space RBMO(μ) 

分 类 号：O177[理学—数学]