Schatten Classes and Commutators of Riesz Transform on Heisenberg Group and Applications

Abstract

We study commutators with the Riesz transforms on the Heisenberg group {mathbb {H}}^{n}. The Schatten norm of these commutators is characterized in terms of Besov norms of the symbol. This generalizes the classical Euclidean results of Peller, Janson–Wolff and Rochberg–Semmes. The method in proof bypasses the use of Fourier analysis, allowing us to address not just the Riesz transforms, but also the Cauchy–Szegő projection and second order Riesz transforms on {mathbb {H}}^{n} among other settings.