CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (6): 1779-1788.

CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES

于林, 王汝慧, 赵守江

School of Science, China Three Gorges University, Yichang 443002, China

收稿日期:2017-07-21 修回日期:2018-01-08 出版日期:2018-12-25 发布日期:2018-12-28
通讯作者: Lin YU E-mail:yulin@ctgu.edu.cn
作者简介:Ruhui WANG,E-mail:rhwang93@163.com;Shoujiang ZHAO,E-mail:shjzhao@163.com
基金资助:
This work was supported by National Natural Science Foundation of China (11601267).

CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES

Lin YU, Ruhui WANG, Shoujiang ZHAO

School of Science, China Three Gorges University, Yichang 443002, China

Received:2017-07-21 Revised:2018-01-08 Online:2018-12-25 Published:2018-12-28
Contact: Lin YU E-mail:yulin@ctgu.edu.cn
Supported by:
This work was supported by National Natural Science Foundation of China (11601267).

摘要/Abstract

摘要： In this paper, the so-called (p, φ)-Carleson measure is introduced and the relationship between vector-valued martingales in the general Campanato spaces L_p,φ(X) and the (p, φ)-Carleson measures is investigated. Specifically, it is proved that for q ∈[2, ∞), the measure dμ:=||df_k||^qdP ⊗ dm is a (q, φ)-Carleson measure on Ω×N for every f ∈ L_q,φ(X) if and only if X has an equivalent norm which is q-uniformly convex; while for p ∈ (1, 2], the measure dμ:=||df_k||^pdP⊗dm is a (p, φ)-Carleson measure on Ω×N implies that f ∈ L_p,φ(X) if and only if X admits an equivalent norm which is p-uniformly smooth. This result extends an earlier result in the literature from BMO spaces to general Campanato spaces.

关键词: Carleson measures, BMO martingales, generalized Campanato spaces, uniformly convex (smooth) Banach spaces

Abstract: In this paper, the so-called (p, φ)-Carleson measure is introduced and the relationship between vector-valued martingales in the general Campanato spaces L_p,φ(X) and the (p, φ)-Carleson measures is investigated. Specifically, it is proved that for q ∈[2, ∞), the measure dμ:=||df_k||^qdP ⊗ dm is a (q, φ)-Carleson measure on Ω×N for every f ∈ L_q,φ(X) if and only if X has an equivalent norm which is q-uniformly convex; while for p ∈ (1, 2], the measure dμ:=||df_k||^pdP⊗dm is a (p, φ)-Carleson measure on Ω×N implies that f ∈ L_p,φ(X) if and only if X admits an equivalent norm which is p-uniformly smooth. This result extends an earlier result in the literature from BMO spaces to general Campanato spaces.

Key words: Carleson measures, BMO martingales, generalized Campanato spaces, uniformly convex (smooth) Banach spaces

于林, 王汝慧, 赵守江. CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES[J]. 数学物理学报(英文版), 2018, 38(6): 1779-1788.

Lin YU, Ruhui WANG, Shoujiang ZHAO. CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES[J]. Acta mathematica scientia,Series B, 2018, 38(6): 1779-1788.

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CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES

CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES

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