Compactness of the Complex Green Operator on C 1 Pseudoconvex Boundaries in Stein Manifolds

Abdullah Alahmari, Emad Solouma, Marin Marin (), A. F. Aljohani and Sayed Saber
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Abdullah Alahmari: Department of Mathematics, Faculty of Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia
Emad Solouma: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
Marin Marin: Department of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Brasov, Romania
A. F. Aljohani: Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71411, Saudi Arabia
Sayed Saber: Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 2722165, Egypt

Mathematics, 2025, vol. 13, issue 21, 1-25

Abstract: We study compactness for the complex Green operator G q 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 C 1 boundary. For 1 ≤ q ≤ n − 2 , we first prove a compactness theorem under weak potential-theoretic hypotheses: if b Ω satisfies weak ( P q ) and weak ( P n − 1 − q ) , then G q and G n − 1 − q are compact on L p , q 2 ( b Ω ) . This extends known C ∞ results in C n to the minimal regularity C 1 and to the Stein setting. On locally convexifiable C 1 boundaries, we obtain a full characterization: compactness of G q is equivalent to simultaneous compactness of G q and G n − 1 − q , to compactness of the ∂ ¯ -Neumann operators N q and N n − 1 − q in the interior, to weak ( P q ) and ( P n − 1 − q ) , and to the absence of (germs of) complex varieties of dimensions q and n − 1 − q on b Ω . A key ingredient is an annulus compactness transfer on Ω + = Ω 2 ∖ Ω 1 ¯ , which yields compactness of N q Ω + 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 L 2 Hodge decompositions, trace-class heat flow, stability under C 1 boundary perturbations, vanishing essential norms, Sobolev mapping (and gains under subellipticity), and compactness of Bergman-type commutators when q = 1 .

Keywords: complex green operator; compactness; boundary value problems; functional analysis; pseudoconvexity; harmonic forms; Stein manifolds (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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