Schatten Properties of Calder\'{o}n--Zygmund Singular Integral Commutator on stratified Lie groups

Abstract

We provide full characterisation of the Schatten properties of $[M_b,T]$, the commutator of Calder\'{o}n--Zygmund singular integral $T$ with symbol $b$ $(M_bf(x):=b(x)f(x))$ on stratified Lie groups $\mathbb{G}$. We show that, when $p$ is larger than the homogeneous dimension $\mathbb{Q}$ of $\mathbb{G}$, the Schatten $\mathcal{L}_p$ norm of the commutator is equivalent to the Besov semi-norm $B_{p}^{\frac{\mathbb{Q}}{p}}$ of the function $b$; but when $p\leq \mathbb{Q}$, the commutator belongs to $\mathcal{L}_p$ if and only if $b$ is a constant. For the endpoint case at the critical index $p=\mathbb{Q}$, we further show that the Schatten $\mathcal{L}_{\mathbb{Q},\infty}$ norm of the commutator is equivalent to the Sobolev norm $W^{1,\mathbb{Q}}$ of $b$. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond.