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

Abstract

Let $P_{\gamma }$ be the orthogonal projection from the space $L^2({\mathbb{B}}_n,d{v}_{\gamma })$ to the standard weighted Bergman space $L_a^2({\mathbb{B}}_n,d{v}_{\gamma })$. 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 Möbius invariant measure on ${\mathbb{B}}_n.$ This answers a question recently raised by K. Zhu.

v1: 33 pages, 5 figures, v2: 35 pages, 5 figures, revisions made, to appear in Indiana Univ. Math. J