Schatten properties of Calderon-Zygmund singular integral commutator on stratified Lie groups

We provide full characterization of the Schatten properties of [M-b,T], the commutator of Calderon-Zygmund singular integral T with symbol b (M(b)f(x):=b(x)f(x)) on stratified Lie groups G. We show that, when p is larger than the homogeneous dimension Q of G, the Schatten L-p norm of the commutator is equivalent to the Besov semi-norm B-p(Q/p) of the function b; but when p <= Q, the commutator belongs to L-p if and only if b is a constant. For the endpoint case at the critical index p = Q, we further show that the Schatten L-Q,L-infinity norm of the commutator is equivalent to the Sobolev norm W-1,W-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. Crown Copyright (C) 2024 Published by Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.