The \bar{\partial}_b Neumann problem on noncharacteristic domains

Abstract

We study the $\bar{\partial}_b$-Neumann problem for domains $\Omega$ 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 L^2, we prove sharp regularity and estimates for solutions. We establish a condition on the boundary which is sufficient for the Kohn Laplacian to be Fredholm on $L^2_{(0,q)}(\Omega)$ and show that this condition always holds when M is embedded as a hypersurface in C^{n+1}. We present examples where the inhomogeneous $\bar{\partial}_b$ equation can always be solved smoothly up to the boundary on (p,q)-forms with 0<q<n-1.