The∂b-Neumann problem on noncharacteristic domains

Abstract

We study the ∂ b -Neumann problem for domains Ω contained in a strictly pseudoconvex manifold M 2n+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 that is sufficient for ? b to be Fredholm on L 2 (0,q) (Ω) and show that this condition always holds when M is embedded as a hypersurface in ℂ n+1 . We present examples where the inhomogeneous ∂ b equation can always be solved in ℂ ∞ (Ω) on (p, q)-forms with1 ≤ q ≤ n - 2.