Schatten classes and commutators of Riesz transform on Heisenberg group and applications

Abstract

We establish the necessary and sufficient conditions for those symbols b on the Heisenberg group H n for which the commutator with the Riesz transform is of Schatten class. Our main result generalises classical results of Peller, Janson-Wolff and Rochberg-Semmes, which address the same question in the Euclidean setting. Moreover, the approach that we develop bypasses the use of Fourier analysis, and can be applied to characterise that the commutator is of the Schatten class in other settings beyond Euclidean.

Fundamental properties of Cauchy--Szegő projection on quaternionic Siegel upper half space and applications

“…A similar Janson-Wolff phenomenon has been demonstrated by Feldman and Rochberg [32] for the Cauchy-Szegő projection on the unit ball and Heisenberg group via the Hankel operators. Here we establish the analogous result of Feldman and Rochberg in our setting, based on our main result, Theorem 1.3, and on the recent breakthrough of Fan, Lacey and Li [28], since the theory of Fourier analysis and Hankel operator is not as effective as in the Euclidean setting or the classical Heisenberg group setting.…”

“…Based on our fundamental results in Lemmas 6.1-6.4, (2) holds by using the argument in [28]. Hence, the proof of Theorem 1.6 is complete.…”