∂̄b-NEUMANN PROBLEM ON NONCHARACTERISTIC DOMAINS

We study the ∂̄b-Neumann problem for domains Ω contained in a strictly pseudoconvex manifold M2n+1 whose boundaries are noncharacteristic and have defining functions depending solely on the real and imaginary parts of a single CR function w. When the Kohn Laplacian is a priori known to have closed range in L2, we prove sharp regularity and estimates for solutions. We establish a condition on the boundary ∂Ω which is sufficient for b to be Fredholm on L2 (0,q) (Ω) and show that this condition always holds when M is embedded as a hypersurface in Cn+1. We present examples where the inhomogenous ∂̄b equation can always be solved in C ∞(Ω) on (p, q)-forms with 1 ≤ q ≤ n − 2.