IUMJ

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

Authors: Joshua Isralowitz

Issue: Volume 62 (2013), Issue 1, 201-233

Abstract:

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