Optimal estimates for martingale transforms

Jinghao Huang; Fedor Sukochev; Dmitriy Zanin

doi:10.1016/j.jfa.2022.109451

Optimal estimates for martingale transforms

Jinghao Huang^*
, Fedor Sukochev
, Dmitriy Zanin

^*Corresponding author for this work

University of New South Wales

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

6 Scopus citations

Abstract

Let (M_m)_m≥0 be an arbitrary filtration in a (non-commutative) probability space (M,τ). Let L₁(M,τ) be the predual of M. For m≥0, we denote by E_{M_m} the conditional expectation from M onto M_m. For a given sequence of signs ϵ={ϵ_m}_m≥0, we define a martingale transform T_ϵx=∑m≥0ϵ_m⋅(E_{M_m}x−E_{M_m−1}x). We present the sharp estimate of distributional function of T_ϵx, ∀x∈L₁(M,τ), in terms of the classical Calderon operator. Our result complements and extends classical results due to Pisier and Xu [42], and Randrianantoanina [44].

Original language	English
Article number	109451
Journal	Journal of Functional Analysis
Volume	282
Issue number	11
DOIs	https://doi.org/10.1016/j.jfa.2022.109451
State	Published - 1 Jun 2022
Externally published	Yes

Keywords

Calderon operator
Hilbert transform
Martingale transforms
Riesz projection

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10.1016/j.jfa.2022.109451

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title = "Optimal estimates for martingale transforms",

abstract = "Let (Mm)m≥0 be an arbitrary filtration in a (non-commutative) probability space (M,τ). Let L1(M,τ) be the predual of M. For m≥0, we denote by EMm the conditional expectation from M onto Mm. For a given sequence of signs ϵ=\{ϵm\}m≥0, we define a martingale transform Tϵx=∑m≥0ϵm⋅(EMmx−EMm−1x). We present the sharp estimate of distributional function of Tϵx, ∀x∈L1(M,τ), in terms of the classical Calderon operator. Our result complements and extends classical results due to Pisier and Xu [42], and Randrianantoanina [44].",

keywords = "Calderon operator, Hilbert transform, Martingale transforms, Riesz projection",

author = "Jinghao Huang and Fedor Sukochev and Dmitriy Zanin",

note = "Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

year = "2022",

month = jun,

day = "1",

doi = "10.1016/j.jfa.2022.109451",

language = "英语",

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journal = "Journal of Functional Analysis",

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TY - JOUR

T1 - Optimal estimates for martingale transforms

AU - Huang, Jinghao

AU - Sukochev, Fedor

AU - Zanin, Dmitriy

PY - 2022/6/1

Y1 - 2022/6/1

N2 - Let (Mm)m≥0 be an arbitrary filtration in a (non-commutative) probability space (M,τ). Let L1(M,τ) be the predual of M. For m≥0, we denote by EMm the conditional expectation from M onto Mm. For a given sequence of signs ϵ={ϵm}m≥0, we define a martingale transform Tϵx=∑m≥0ϵm⋅(EMmx−EMm−1x). We present the sharp estimate of distributional function of Tϵx, ∀x∈L1(M,τ), in terms of the classical Calderon operator. Our result complements and extends classical results due to Pisier and Xu [42], and Randrianantoanina [44].

AB - Let (Mm)m≥0 be an arbitrary filtration in a (non-commutative) probability space (M,τ). Let L1(M,τ) be the predual of M. For m≥0, we denote by EMm the conditional expectation from M onto Mm. For a given sequence of signs ϵ={ϵm}m≥0, we define a martingale transform Tϵx=∑m≥0ϵm⋅(EMmx−EMm−1x). We present the sharp estimate of distributional function of Tϵx, ∀x∈L1(M,τ), in terms of the classical Calderon operator. Our result complements and extends classical results due to Pisier and Xu [42], and Randrianantoanina [44].

KW - Calderon operator

KW - Hilbert transform

KW - Martingale transforms

KW - Riesz projection

UR - https://www.scopus.com/pages/publications/85125361059

U2 - 10.1016/j.jfa.2022.109451

DO - 10.1016/j.jfa.2022.109451

M3 - 文章

AN - SCOPUS:85125361059

SN - 0022-1236

VL - 282

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 11

M1 - 109451

ER -

Optimal estimates for martingale transforms

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