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

TL;DR

This paper characterizes when certain commutators on weighted Bergman spaces belong to Schatten p-classes, linking this property to the mean oscillation of functions, thus answering a question posed by K. Zhu.

Contribution

It provides a complete characterization of Schatten p-class membership for commutators on weighted Bergman spaces, connecting operator theory with function mean oscillation.

Findings

Commutator belongs to Schatten p-class iff mean oscillation is in L^p.

The characterization holds for p > 2n/(n+1+γ).

Answers a recent question by K. Zhu.

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.