The ∂_b-Neumann problem on noncharacteristic domains

Robert K. Hladky

Vol. 242 (2009), No. 1, 71–112

DOI: 10.2140/pjm.2009.242.71

Abstract

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 that is sufficient for □_b to be Fredholm on L_(0,q)²(Ω) and show that this condition always holds when M is embedded as a hypersurface in ℂⁿ⁺¹. We present examples where the inhomogeneous ∂_b equation can always be solved in C^∞(Ω) on (p,q)-forms with 1 ≤ q ≤ n − 2.

Keywords

CR manifold, Kohn Laplacian, subelliptic, boundary regularity, tangential Cauchy–Riemann equation

Mathematical Subject Classification 2000

Primary: 32V10, 32V20, 32V40, 32W10

Secondary: 35H20

Milestones

Received: 23 September 2008

Revised: 12 February 2009

Accepted: 6 April 2009

Published: 1 September 2009