Optimal domain class for the fixed-radius averaging criterion

Determine the optimal class of bounded complex domains on which fixed-radius Euclidean-volume averages over Bergman geodesic balls characterize compactness of bounded-symbol Toeplitz operators, specifically including whether the criterion holds on smooth weakly pseudoconvex domains.

Background

The paper proves that, on bounded strongly pseudoconvex domains with $C^2$ boundary, a bounded-symbol Toeplitz operator is compact exactly when its Euclidean-volume averages over Bergman geodesic balls of any fixed positive radius vanish at the boundary. It also establishes that the result remains valid under the stated $C^2$ regularity, even without a positive Hölder modulus for the second boundary derivatives.

The authors explicitly note that the optimal domain class beyond the strongly pseudoconvex setting has not been identified. Although the polydisc counterexample rules out an extension to all bounded pseudoconvex domains, it does not resolve the intermediate case of smooth weakly pseudoconvex domains. The open problem is therefore to determine the sharp geometric and boundary-regularity hypotheses under which the fixed-radius averaging criterion remains valid.