Compactness of Riesz transform commutator on stratified Lie groups

TL;DR

This paper characterizes when the commutators of Riesz transforms on stratified Lie groups are compact, using a new geometric construction related to the kernel's lower bounds and functions in VMO.

Contribution

It provides a concrete geometric construction and a characterization of compactness for Riesz transform commutators on stratified Lie groups, extending previous results to a broader setting.

Findings

Characterization of compactness of Riesz transform commutators

Construction of twisted truncated sector related to kernel bounds

Extension of VMO function space analysis on stratified Lie groups

Abstract

Let $\mathcal G$ be a stratified Lie group and $\{\X_j\}_{1 \leq j \leq n}$ a basis for the left-invariant vector fields of degree one on $\mathcal G$. Let $\Delta = \sum_{j = 1}^n \X_j^2 $ be the sub-Laplacian on $\mathcal G$. The $j^{\mathrm{th}}$ Riesz transform on $\mathcal G$ is defined by $R_j:= \X_j (-\Delta)^{-\frac{1}{2}}$, $1 \leq j \leq n$. 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 $\mathcal G$. Then we obtain the characterisation of compactness of the commutators of $R_j$ with a function $b\in$ VMO$(\mathcal G)$, the space of functions with vanishing mean oscillation on $\mathcal G$.