Compactness of Riesz transform commutator on stratified Lie groups

Peng Chen; Xuan Thinh Duong; Ji Li; Qingyan Wu

doi:10.1016/j.jfa.2019.05.008

Compactness of Riesz transform commutator on stratified Lie groups

Peng Chen, Xuan Thinh Duong, Ji Li, Qingyan Wu^*

^*Corresponding author for this work

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

42 Citations (Scopus)

Abstract

Let G be a stratified Lie group and {X_j}_1≤j≤n a basis for the left-invariant vector fields of degree one on G. Let Δ=∑_j=1 ⁿX_j ² be the sub-Laplacian on G. The jth Riesz transform on G is defined by R_j:=X_j(−Δ)^-1/2, 1≤j≤n. In this paper, we provide a concrete construction of the "twisted truncated sector" which is related to the pointwise lower bound of the kernel of R_j on G. Then we obtain the characterisation of compactness of the commutators of R_j with a function b∈VMO(G), the space of functions with vanishing mean oscillation on G.

Original language	English
Pages (from-to)	1639-1676
Number of pages	38
Journal	Journal of Functional Analysis
Volume	277
Issue number	6
DOIs	https://doi.org/10.1016/j.jfa.2019.05.008
Publication status	Published - 15 Sept 2019

Keywords

Stratified Lie groups
Riesz transforms
VMO space
Commutator

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10.1016/j.jfa.2019.05.008

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title = "Compactness of Riesz transform commutator on stratified Lie groups",

abstract = "Let G be a stratified Lie group and {Xj}1≤j≤n a basis for the left-invariant vector fields of degree one on G. Let Δ=∑j=1 nXj 2 be the sub-Laplacian on G. The jth Riesz transform on G is defined by Rj:=Xj(−Δ)-1/2, 1≤j≤n. In this paper, we provide a concrete construction of the {"}twisted truncated sector{"} which is related to the pointwise lower bound of the kernel of Rj on G. Then we obtain the characterisation of compactness of the commutators of Rj with a function b∈VMO(G), the space of functions with vanishing mean oscillation on G.",

keywords = "Stratified Lie groups, Riesz transforms, VMO space, Commutator",

author = "Peng Chen and Duong, {Xuan Thinh} and Ji Li and Qingyan Wu",

year = "2019",

month = sep,

day = "15",

doi = "10.1016/j.jfa.2019.05.008",

language = "English",

volume = "277",

pages = "1639--1676",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Elsevier",

number = "6",

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TY - JOUR

T1 - Compactness of Riesz transform commutator on stratified Lie groups

AU - Chen, Peng

AU - Duong, Xuan Thinh

AU - Li, Ji

AU - Wu, Qingyan

PY - 2019/9/15

Y1 - 2019/9/15

N2 - Let G be a stratified Lie group and {Xj}1≤j≤n a basis for the left-invariant vector fields of degree one on G. Let Δ=∑j=1 nXj 2 be the sub-Laplacian on G. The jth Riesz transform on G is defined by Rj:=Xj(−Δ)-1/2, 1≤j≤n. In this paper, we provide a concrete construction of the "twisted truncated sector" which is related to the pointwise lower bound of the kernel of Rj on G. Then we obtain the characterisation of compactness of the commutators of Rj with a function b∈VMO(G), the space of functions with vanishing mean oscillation on G.

AB - Let G be a stratified Lie group and {Xj}1≤j≤n a basis for the left-invariant vector fields of degree one on G. Let Δ=∑j=1 nXj 2 be the sub-Laplacian on G. The jth Riesz transform on G is defined by Rj:=Xj(−Δ)-1/2, 1≤j≤n. In this paper, we provide a concrete construction of the "twisted truncated sector" which is related to the pointwise lower bound of the kernel of Rj on G. Then we obtain the characterisation of compactness of the commutators of Rj with a function b∈VMO(G), the space of functions with vanishing mean oscillation on G.

KW - Stratified Lie groups

KW - Riesz transforms

KW - VMO space

KW - Commutator

UR - http://www.scopus.com/inward/record.url?scp=85066469541&partnerID=8YFLogxK

UR - http://purl.org/au-research/grants/arc/DP160100153

U2 - 10.1016/j.jfa.2019.05.008

DO - 10.1016/j.jfa.2019.05.008

M3 - Article

AN - SCOPUS:85066469541

SN - 0022-1236

VL - 277

SP - 1639

EP - 1676

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 6

ER -

Compactness of Riesz transform commutator on stratified Lie groups

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Multiparameter Harmonic Analysis: Weighted Estimates for Singular Integrals

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Compactness of Riesz transform commutator on stratified Lie groups

Abstract

Keywords

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Multiparameter Harmonic Analysis: Weighted Estimates for Singular Integrals

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