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

TL;DR

This paper investigates the regularity and solvability of the ar_b-Neumann problem on certain noncharacteristic domains within strictly pseudoconvex manifolds, providing conditions for Fredholm properties and smooth solutions.

Contribution

It establishes sharp regularity estimates, a boundary condition for the Fredholm property of the Kohn Laplacian, and demonstrates smooth solvability of the ar_b equation in specific cases.

Findings

Proves sharp regularity and estimates for solutions when the Kohn Laplacian has closed range.

Identifies a boundary condition sufficient for the Kohn Laplacian to be Fredholm on L^2 spaces.

Shows examples where the ar_b equation can be solved smoothly up to the boundary.

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.