Continuous boundary values of analytic functions of several complex variables

Abstract

Our method is elementary, and was suggested by the proofs of the Hartogs' Extension Theorem given by Bochner [I] and Martinelli [6]. We remark that the differential forms co appearing in the theorem are a natural generalization to several variables of the analytic differentials which play the corresponding role in the well-known one-variable theorem. By appealing to the regularity theorem of Kohn [4] for the 5-Neumann problem we obtain as a corollary that if the Levi form on r has at least one positive eigenvalue at each point, thenfEA (r) if and only if f is a weak solution of the tangential Cauchy-Riemann equations. (With this hypothesis on r, it follows that if D is connected, then so is r.) This theorem was obtained by Fichera [2] under the additional hypotheses that r is connected and f is the boundary value of a function with finite Dirichlet integral. His proof is based on an approximation theorem for harmonic functions in. the Dirichlet norm. Kohn and Rossi [5] have shown that the theorem holds for CI functions when D is a finite domain on a complex manifold and the Levi form satisfies the above condition on r. They also obtain extension theorems for (p, q) forms.