Compactness of the Complex Green Operator on C1 Pseudoconvex Boundaries in Stein Manifolds

We study compactness for the complex Green operator Gq associated with the Kohn Laplacian □b on boundaries of pseudoconvex domains in Stein manifolds. Let Ω⋐X be a bounded pseudoconvex domain in a Stein manifold X of complex dimension n with C1 boundary. For 1≤q≤n−2, we first prove a compactness theorem under weak potential-theoretic hypotheses: if bΩ satisfies weak (Pq) and weak (Pn−1−q), then Gq and Gn−1−q are compact on Lp,q2(bΩ). This extends known C∞ results in Cn to the minimal regularity C1 and to the Stein setting. On locally convexifiable C1 boundaries, we obtain a full characterization: compactness of Gq is equivalent to simultaneous compactness of Gq and Gn−1−q, to compactness of the ∂¯-Neumann operators Nq and Nn−1−q in the interior, to weak (Pq) and (Pn−1−q), and to the absence of (germs of) complex varieties of dimensions q and n−1−q on bΩ. A key ingredient is an annulus compactness transfer on Ω+=Ω2∖Ω1¯, which yields compactness of NqΩ+ from weak (P) near each boundary component and allows us to build compact ∂¯b-solution operators via jump formulas. Consequences include the following: compact canonical solution operators for ∂¯b, compact resolvent for □b on the orthogonal complement of its harmonic space (hence discrete spectrum and finite-dimensional harmonic forms), equivalence between compactness and standard compactness estimates, closed range and L2 Hodge decompositions, trace-class heat flow, stability under C1 boundary perturbations, vanishing essential norms, Sobolev mapping (and gains under subellipticity), and compactness of Bergman-type commutators when q=1.