A boundary Morera theorem

Abstract

LetD ⊂C ^N,N ≥ 2 be a bounded open set withC ² boundary and letL be an open connected set of affine complex hyperplanes inC ^N containing a hyperplane that misses$\bar D$. LetE = ∪_Λ∈LΛ, Γ =E ∩bD. Suppose thatf ∈C(Γ) and assume that

$$\int_{\Lambda \cap bD} {f\alpha = 0} $$

for every Λ ∈L that meetsbD transversely and for every (N −1,N −2)-form with constant coefficients. Thenf satisfies the weak tangential Cauchy-Riemann equations on Γ. It is shown by an example that the condition thatL contains a hyperplane that misses$\bar D$ cannot be dropped.