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Formules explicites pour les solutions minimales de l’équation ∂ ¯ u = f dans la boule et dans le polydisque de ℂ n

Philippe Charpentier (1980)

Annales de l'institut Fourier

Dans cet article, on construit tout d’abord un noyau de Cauchy explicite dans la boule unité B de C n dont les valeurs au bord sont égales au noyau de Szegö. Puis, à partir de ce noyau, on construit explicitement les noyaux qui fournissent les solutions de l’équation ∂ ‾ u = f qui sont orthogonales aux fonctions holomorphes dans les espaces L 2 ( d σ α ) , où d σ α ( z ) = ( 1 - | z | 2 ) d λ ( z ) , d λ ( z ) étant la mesure de Lebesgue et α un réel > - 1 . Nous donnons ensuite les principales estimations dedans et au bord que vérifient ces solutions. Dans une deuxième...

Fundamental solutions of the complex Monge-Ampère equation

Halil Ibrahim Celik, Evgeny A. Poletsky (1997)

Annales Polonici Mathematici

We prove that any positive function on ℂℙ¹ which is constant outside a countable G δ -set is the order function of a fundamental solution of the complex Monge-Ampère equation on the unit ball in ℂ² with a singularity at the origin.

Geometric conditions which imply compactness of the ∂ ¯ -Neumann operator

Emil Straube (2004)

Annales de l’institut Fourier

For smooth bounded pseudoconvex domains in ℂ 2 , we provide geometric conditions on the boundary which imply compactness of the ∂ ¯ -Neumann operator. It is noteworthy that the proof of compactness does not proceed via verifying the known potential theoretic sufficient conditions.

Hartogs theorem for forms : solvability of Cauchy-Riemann operator at critical degree

Chin-Huei Chang, Hsuan-Pei Lee (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

The Hartogs Theorem for holomorphic functions is generalized in two settings: a CR version (Theorem 1.2) and a corresponding theorem based on it for C k ∂ ¯ -closed forms at the critical degree, 0 ≤ k ≤ ∞ (Theorem 1.1). Part of Frenkel’s lemma in C k category is also...

Henkin-Ramirez formulas with weight factors

B. Berndtsson, Mats Andersson (1982)

Annales de l'institut Fourier

We construct a generalization of the Henkin-Ramírez (or Cauchy-Leray) kernels for the ∂ ‾ -equation. The generalization consists in multiplication by a weight factor and addition of suitable lower order terms, and is found via a representation as an “oscillating integral”. As special cases we consider weights which behave like a power of the distance to the boundary, like exp- ϕ with ϕ convex, and weights of polynomial decrease in C n . We also briefly consider kernels with singularities on subvarieties...

Hölder a priori estimates for second order tangential operators on CR manifolds

Annamaria Montanari (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

On a real hypersurface M in ℂ n + 1 of class C 2 , α we consider a local CR structure by choosing n complex vector fields W j in the complex tangent space. Their real and imaginary parts span a 2 n -dimensional subspace of the real tangent space, which has dimension 2 n + 1 . If the Levi matrix of M is different from zero at every point, then we can generate the missing direction. Under this assumption we prove interior a priori estimates of Schauder type for solutions of a class of second order partial differential equations...

Hölder and Lp estimates for the solutions of the ∂-equation in non-smooth strictly pseudoconvex domains.

Josep M. Burgués Badía (1990)

Publicacions Matemàtiques

Let D be a bounded strict pseudoconvex non-smooth domain in Cn. In this paper we prove that the estimates in Lp and Lipschitz classes for the solutions of the ∂-equation with Lp-data in regular strictly pseudoconvex domains (see [2]) are also valid for D. We also give estimates of the same type for the ∂b in the regular part of the boundary of these domains.

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