Schatten $p$ class commutators on the weighted Bergman space $L^2_a (\mathbb{B}_n, dv_\gamma)$ for $\frac{2n}{n + 1 + \gamma} < p < \infty$

Abstract

Let $P_\gamma$ be the orthogonal projection from the space $L ^2 (\mathbb{B}_n, dv_\gamma)$ to the standard weighted Bergman space $L_a ^2 (\mathbb{B}_n, dv_\gamma)$. In this paper, we characterize the Schatten $p$ class membership of the commutator $[M_f, P_\gamma]$ when $\frac{2n}{n + 1 + \gamma} < p < \infty$. In particular, if $\frac{2n}{n + 1 + \gamma} < p < \infty$, then we show that $[M_f, P_\gamma]$ is in the Schatten $p$ class if and only if the mean oscillation MO${}_\gamma (f)$ is in $ L^p(\mathbb{B}_n, d\zeta)$ where $d\zeta$ is the M\"{o}bius invariant measure on $\mathbb{B}_n.$ This answers a question recently raised by K. Zhu.