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

TL;DR

This paper characterizes when the commutator of the Bergman projection is bounded or compact on strongly pseudoconvex domains with minimal smoothness, linking it to BMO and VMO function spaces.

Contribution

It provides a new characterization of the boundedness and compactness of the commutator in terms of BMO and VMO conditions on the symbol function, extending previous results to less smooth domains.

Findings

Boundedness of the commutator characterized by BMO conditions.

Compactness of the commutator characterized by VMO conditions.

Established equivalence of BMO/VMO with other known function spaces.

Abstract

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.