Extension of CR-functions

Abstract

Let Γ be a real hypersurface in ¢ⁿ, n ≥ 2, oriented of class C¹, compact, connected with boundary ϖΓ. Suppose that ϖΓ belongs to a C^∞-hypersurface M and there exists a relatively open subset A of M such that ϖA = ϖΓ, and that Γ ∩ M = ϖΓ. Under these hypotheses, in a previous paper [2] G. Lupacciolu and the author have proved that if M is the zero-set of a pluriharmonic function on a neighbourhood of $\bar D$, every Lipschitz continuous CR-function f on Γ^o = Γ \ ϖΓ extends uniquely by a function F, holomorphic on D, where D is an open set of ¢ⁿ having Γ ∪ A as its boundary. Moreover, F is continuous on D ∪ Γ^o. The present paper aims at generalizing this result to the case when M is Levi-flat or Levi-pseudoconcave with respect to D. The positive answer is obtained in two particular cases (Theorems 1 and 3).