On the weak* convergence in HF                             1(Rn)

Chin Cheng Lin

doi:10.1016/j.jmaa.2017.06.005

On the weak* convergence in H_F ¹(Rⁿ)

Chin Cheng Lin

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let VMO_F denote the closure of C_c∩Lip with respect to the seminorm ‖⋅‖_{BMO_F}, where F is a family of sections and the space BMO_F 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 VMO_F is the Hardy space H_F 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 H_F ¹ to a function in L¹(dμ) implies the weak* convergence.

Original language	English
Pages (from-to)	463-476
Number of pages	14
Journal	Journal of Mathematical Analysis and Applications
Volume	455
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2017.06.005
State	Published - 1 Nov 2017

Keywords

A weights
BMO
Hardy spaces
Monge–Ampère equation
VMO

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10.1016/j.jmaa.2017.06.005

Cite this

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title = "On the weak* convergence in HF 1(Rn)",

abstract = "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 Guti{\'e}rrez (1996) [4]. Sections play an important role in the investigation of Monge–Amp{\`e}re equation and the linearized Monge–Amp{\`e}re 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.",

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T1 - On the weak* convergence in HF 1(Rn)

AU - Lin, Chin Cheng

PY - 2017/11/1

Y1 - 2017/11/1

N2 - 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.

AB - 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.

KW - A weights

KW - BMO

KW - Hardy spaces

KW - Monge–Ampère equation

KW - VMO

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DO - 10.1016/j.jmaa.2017.06.005

M3 - 期刊論文

AN - SCOPUS:85020468321

SN - 0022-247X

VL - 455

SP - 463

EP - 476

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -

On the weak* convergence in HF 1(Rn)

Abstract

Keywords

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On the weak* convergence in H_F ¹(Rⁿ)