On the weak* convergence in HF 1(Rn)

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Let VMOF denote the closure of Cc∩Lip with respect to the seminorm ‖⋅‖BMOF, where F is a family of sections and the space BMOF associated to the family F was introduced by Caffarelli and Gutiérrez (1996) [4]. Sections play an important role in the investigation of Monge–Ampère equation and the linearized Monge–Ampère equation. We show that the dual of VMOF is the Hardy space HF 1 defined in Ding and Lin (2005) [8]. As an application, we prove that μ-almost everywhere convergence of a sequence of functions bounded in HF 1 to a function in L1(dμ) implies the weak* convergence.

Original languageEnglish
Pages (from-to)463-476
Number of pages14
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - 1 Nov 2017


  • A weights
  • BMO
  • Hardy spaces
  • Monge–Ampère equation
  • VMO


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