Let Ω ⊂ ℂm × ℝn be an open subset with smooth boundary. The de Rham complex of ℝn and the Dolbeaut complex of ℂm induce a natural elliptic complex on Ω D : Λp(Ω) → Λp+1(Ω), p = 0, . . . , m + n. D induces a natural involutive (formally integrable) structure, Db, on the boundary of Ω. Let Σ ⊂ ∂Ω be the set of points where Db is not elliptic. We assume that the Levi form is positive definite at each point of Σ. Away from Σ we make no assumption. Under these conditions we show that the cohomology for D vanishes in dimension p ≥ 1.
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