The Kohn-Laplace equation on abstract CR manifolds: Global regularity

Abstract

Let M M be a compact, pseudoconvex-oriented, ( 2 n + 1 ) (2n+1) -dimensional, abstract CR manifold of hypersurface type, n ≥ 2 n\geq 2 . We prove the following: (i) If M M admits a strictly CR-plurisubharmonic function on ( 0 , q 0 ) (0,q_0) -forms, then the complex Green operator G q G_q exists and is continuous on L 0 , q 2 ( M ) L^2_{0,q}(M) for degrees q 0 ≤ q ≤ n − q 0 q_0\le q\le n-q_0 . In the case that q 0 = 1 q_0=1 , we also establish continuity for G 0 G_0 and G n G_n . Additionally, the ∂ ¯ b \bar {\partial }_{b} -equation on M M can be solved in C ∞ ( M ) C^\infty (M) . (ii) If M M satisfies “a weak compactness property” on ( 0 , q 0 ) (0,q_0) -forms, then G q G_q is a continuous operator on H 0 , q s ( M ) H^s_{0,q}(M) and is therefore globally regular on M M for degrees q 0 ≤ q ≤ n − q 0 q_0\le q\le n-q_0 ; and also for the top degrees q = 0 q=0 and q = n q=n in the case q 0 = 1 q_0=1 . We also introduce the notion of a “plurisubharmonic CR manifold” and show that it generalizes the notion of “plurisubharmonic defining function” for a domain in C N \mathbb {C}^N and implies that M M satisfies the weak compactness property.