Noyau et métrique de Bergman dans des formules de représentations pour les convexes de type fini et applications

Abstract : In strictly pseudoconvex domains, S. G. Krantz found a solution for the Cauchy-Riemann equation for a bounded data in the Lipschitz space $\Lambda^(\frac(1)(2))$. Recently, a similar result was obtained in convex domains of finite type m by A. Cumenge and B. Fischer, J. E. Fornaess, K. Diederich : for a bounded data, the solution belongs to $\Lambda^(\frac(1)(m))$. However, S. G. Krantz's result in strictly pseudoconvex domains was improved by P. Greiner and E. Stein. Under the same hypothesis, they obtained a solution for the Cauchy-Riemann equation in the anisotropic Lispchitz space $\Lambda^(\frac(1)(2), 1)$. So, it seems that there is a better regularity for the solution in tangent directions. Our work consists in finding optimal lipschitzian estimates in a convex, bounded and smooth domain of finite type m. In the first part, we use the integral formula builded by A. Cumenge with Berndtsson-Andersson kernel. This construction is "semi-geometric" because the weight depends on Bergman kernel but the section is the classical Bochner-Martinelli. We obtained then a first result which is not optimal. We decided to keep it because it explains the usual approach and the difficulties for estimations. This result will be improved in the third part. In all the results, the data is isotropic and only the solution is anisotropic. So, we thougth interesting to try an approach where the data is bounded with an anisotropic norm. For this, we used the kappa norm introduced by Bruna-Charpentier-Dupain which is a sort of linear version of Kobayashi norm. The solution in then in the isotropic Zygmund space$\Lambda^1(\Omega)$. In the second part of our work, we construct a kernel totally geometric because in both weight and section, only Bergman kernel and Bergman metric are used. This construction is similar to Berndtsson-Andersson ones, but we can't used directly their results because our section is not checking their sufficient hypothesis. We then obtain a representation formula for (p,q)-forms. With this choice for the weight, we are cancelling the boundary term which appears in homotopy formulas and we are directly obtaining a solution for the Cauchy-Riemann equation for "delta-bar" closed forms. In the third part, we are applying this kernel and improving first part's result with an optimal estimate: for a bounded data, we can show that a solution is in an anisotropic space for function : $\Gamma_(\rho)^(\frac(1)(m))(\Omega)$ introduced by J. McNeal and E. Stein. This is a Lipschitz $\frac(1)(m)$ space for an anisotropic metric $\rho$ based on McNeal pseudometric. In order to prove this result, we need an adaptation of "Hardy-Littlewood" lemma and then we are able to estimate almost all the terms in the kernel. For the last one, which contain the maximal singularitie, we can't derive and then we need a direct approach. For this, we give an equivalent definition for $\Gamma_(\rho)^(\frac(1)(m))(\Omega)$ which is based on a sort of anisotropic approximation of unity adapted to the geometry of the domain. In the last part, we are giving a second application : we obtain a new proof for anisotropic Greiner and Stein theorem in strictly pseudoconvex domains. This is quite natural to obtain it because our aim was to obtain this sort of results, but we need here to express it in the euclidian geometry without using $\rho$ metric. So, this result show that we obtain the optimal estimates.
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https://tel.archives-ouvertes.fr/tel-00004225
Contributor : Mathieu Fructus <>
Submitted on : Tuesday, January 20, 2004 - 10:35:28 AM
Last modification on : Friday, January 10, 2020 - 9:08:06 PM
Long-term archiving on: : Wednesday, September 12, 2012 - 12:40:18 PM

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• HAL Id : tel-00004225, version 1

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Mathieu Fructus. Noyau et métrique de Bergman dans des formules de représentations pour les convexes de type fini et applications. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2003. Français. ⟨tel-00004225⟩

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