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Endpoint estimates of linear commutators on Hardy spaces over spaces of homogeneous type
Author(s) -
Liu Liguang,
Chang DerChen,
Fu Xing,
Yang Dachun
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5112
Subject(s) - mathematics , hardy space , commutator , pure mathematics , type (biology) , bounded function , diagonal , measure (data warehouse) , mathematical analysis , space (punctuation) , operator (biology) , algebra over a field , ecology , linguistics , philosophy , lie conformal algebra , geometry , biochemistry , chemistry , repressor , database , computer science , transcription factor , gene , biology
Let ( X , d , μ ) be a metric measure space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors prove that bilinear operators, which are finite combinations of compositions of commutators and Calderón‐Zygmund operators, are bounded from H 1 ( X ) × BMO ( X ) to L 1 ( X ) . The authors also prove that the commutator, generated by any b ∈ BMO ( X ) and Calderón‐Zygmund operator, is bounded from the Hardy‐type spaceH b 1 ( X ) to the Hardy space H 1 ( X ) , whereH b 1 ( X ) is the largest subspace Y of H 1 ( X ) that ensures the boundedness of the commutators from Y to L 1 ( X ) . The novelties appearing in these approaches exist in applications of the multiresolution analysis of the wavelets on metric measure spaces of homogeneous type, the bilinear decomposition of the product space H 1 ( X ) × BMO ( X ) , the (sub)bilinear decomposition of commutators, the proof of off‐diagonal estimates of the action of Calderón‐Zygmund operators on the wavelet functions, and the boundedness of the almost diagonal matrix on the spaces H 1 ( X ) and BMO ( X ) . Notably, throughout this article, μ is not assumed to satisfy the reverse doubling condition.