Poincar� lemma for tangential Cauchy Riemann complexes

Nacinovich, Mauro

doi:10.1007/bf01451852

Cited by 31 publications

(27 citation statements)

References 12 publications

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“…Indeed, several papers are devoted to the interplay between the signature of the Levi form and the possibility of locally solving the system d u = f , possibly in a smaller neighborhood; see Lewy [13], Hörmander [11], Andreotti and Hill [2], Andreotti, Fredricks, and Nacinovich [1], Nacinovich [15,16,17], Kashiwara and Schapira [12], Michel [14], Treves [20,21,22,23], Chen and Shaw [9], Peloso and Ricci [19], Hill and Nacinovich [10], Nicola [18]. We also refer, for other invariants, to the contributions by Cordaro and Hounie [4,5,6], Cordaro and Treves [7,8], Chanillo and Treves [3].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

On local solvability of certain differential complexes

Nicola

2007

Proc. Amer. Math. Soc.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

On local solvability of certain differential complexes

Nicola

2007

Proc. Amer. Math. Soc.

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“…We first recall Barkatou's result This diagram is commutative and by the Poincaré lemma for the ∂ M operator in q-concave CR manifolds [20], both horizontal lines are exact. Moreover the sheaves D p,j and E p,j are sheaves of E-modules and consequently acyclic.…”

Section: Approximation Theoremmentioning

confidence: 92%

Some applications of Serre duality in CR manifolds

Laurent-Thiébaut

Leiterer

1999

Nagoya Mathematical Journal

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Abstract. Applying the methods of Serre duality in the setting of CR manifolds we prove approximation theorems and we study the Hartogs-Bochner phenomenon in 1-concave CR generic manifolds.In this paper we study Serre duality in CR manifolds to get some approximation theorem and a better understanding of the Hartogs-Bochner phenomenon in 1-concave CR manifolds.Because of the lack of the Dolbeault isomorphism for the ∂ b -cohomology in CR manifolds, Serre duality in this setting is not an exact copy of Serre duality in complex manifolds. The main point is that one has to be particularly careful and always to differentiate cohomology with smooth coefficients and current coefficients. Our main theorem in CR-manifolds (Theorem 3.2) is a consequence of rather classical abstract results on duality for complexes of topological vector spaces which we recall in Section 1 (cf.[3], [9], [21]). Note that in the case of compact CR manifolds, Serre duality was already studied by Hill and Nacinovich in [13].Associating Serre duality with Malgrange's theorem on the vanishing of the ∂ b -cohomology in top degree [19] and regularity for the tangential Cauchy-Riemann operator in bidegree (0, 1) [5], we give several applications for 1-concave CR manifolds in Section 4. One of the most interesting is perhaps the study of the Hartogs-Bochner phenomenon. More precisely, we consider a connected non compact C ∞ -smooth 1-concave CR manifold M and a relatively compact open subset D with C ∞ -smooth boundary in M such that M D is connected. The question is : does any smooth CR function f on ∂D extend to a CR function F in D. It turns out that, for the Hartogs-Bochner phenomenon, 1-concave CR manifolds are rather similar to non compact complex manifolds. In both cases the Hartogs-Bochner phenomenon holds if D is sufficiently small, by a result of Henkin [11] in

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“…The above condition is considered by Nacinovich in [17] where existence theorems for the tangential∂ system are derived. Our task is to refine the above criterion and move q to lower values.…”

Section: Q-pseudoconvexity In Higher Codimensionmentioning

confidence: 99%

Non-subelliptic estimates for the tangential Cauchy–Riemann system

2006

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Abstract. We prove non-subelliptic estimates for the tangential Cauchy-Riemann system over a weakly "q-pseudoconvex" higher codimensional submanifold M of C n . Let us point out that our hypotheses do not suffice to guarantee subelliptic estimates, in general. Even more: hypoellipticity of the tangential C-R system is not in question (as shows the example by Kohn of [14] in case of a Levi-flat hypersurface). However our estimates suffice for existence of smooth solutions to the inhomogeneous C-R equations in certain degree.The main ingredients in our proofs are the weighted L 2 estimates by Hörmander [12] and Kohn [14] of §2 and the tangential∂-Neumann operator by Kohn of §4; for this latter we also refer to the book [5]. As for the notion of q pseudoconvexity we follow closely Zampieri [19]. The main technical result, Theorem 2.1, is a version for "perturbed" q-pseudoconvex domains of a similar result by Ahn [1] who generalizes in turn Chen-Shaw [5].

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Poincar� lemma for tangential Cauchy Riemann complexes

Cited by 31 publications

References 12 publications

On local solvability of certain differential complexes

On local solvability of certain differential complexes

Some applications of Serre duality in CR manifolds

Non-subelliptic estimates for the tangential Cauchy–Riemann system

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