Equivalence of property (P_q) and compactness of the ∂-Neumann operator (general case)

Determine whether property (P_q), in the sense of Sibony–Fu–Straube for 1 ≤ q ≤ n, is equivalent to the compactness of the ∂-Neumann operator on general smoothly bounded pseudoconvex domains in C^n, beyond the classes where equivalence is already known (locally convexifiable domains and complete Hartogs domains).

Background

Property (P_q) is a potential-theoretic condition introduced by Catlin and extended by Sibony and Fu–Straube that has strong implications for the ∂-Neumann problem, including compactness of the ∂-Neumann operator and global regularity. The paper reviews extensive applications and known implications of property (P_q).

The equivalence between property (P_q) and compactness has been established for certain classes: locally convexifiable domains (Fu–Straube) and complete Hartogs domains (Christ–Fu). However, the authors note that it remains unresolved in the general setting, making the full equivalence an explicit open problem.