The Commutator of the Bergman Projection on Strongly Pseudoconvex Domains with Minimal Smoothness

Consider a bounded, strongly pseudoconvex domain $D\subset {\mathbb{C}}^n$ with minimal smoothness (namely, the class $C^2$) and let $b$ be a locally integrable function on $D$. We characterize boundedness (resp., compactness) in $L^p(D),p>1$, of the commutator $[b,P]$ of the Bergman projection $P$ in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on $b$. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka.