Maximum modulus sets

Duchamp, Thomas ; Stout, Edgar Lee

Annales de l'Institut Fourier, Tome 31 (1981), p. 37-69 / Harvested from Numdam

Résumé

Nous étudions les sous-ensembles du bord d’un domaine strictement pseudoconvexe $D$ de dimension $N$, où la valeur absolue d’une fonction $f$ de $A(D)$ ou de $A^k(D)$ prend son maximum. Ces ensembles sont les maximum modulus sets du titre. Si $\Sigma \subset bD$ est une variété différentiable de dimension réelle $N$, et si $\Sigma $ est l’ensemble des points où la valeur absolue d’une fonction $f\in A^2(D)$ atteint son maximum, alors $\Sigma $ est totalement réelle et elle admet une structure feuilletée avec comme feuilles des variétés compactes qui sont des ensembles pics d’interpolation. Il y a une converse partielle dans le cas analytique réel. Deux fonctions de $A^2(D)$ qui ont la même variété différentiable de dimension $N$ comme “maximum modulus set”, satisfont une relation analytique, et cette relation est polynomiale si une classe particulière de $H_1(D,{\bf R})$ s’annule ou si $\overline{D} \subset {\bf C}^N$ est polynomialement convexe. Finalement, pour toute fonction $f\in A(D)$, la dimension topologique de l’ensemble des points où $|f|$ prend son maximum est au plus $N$.

Abstract

We investigate some aspects of maximum modulus sets in the boundary of a strictly pseudoconvex domain $D$ of dimension $N$. If $\Sigma \subset bD$ is a smooth manifold of dimension $N$ and a maximum modulus set, then it admits a unique foliation by compact interpolation manifolds. There is a semiglobal converse in the real analytic case. Two functions in $A^2(D)$ with the same smooth $N$-dimensional maximum modulus set are analytically related and are polynomially related if a certain homology class in $H_1(D,{\bf R})$ vanishes or if $\overline{D} \subset {\bf C}^N$ is polynomially convex. Finally, the maximum modulus set of an arbitrary $f\in A(D)$ has dimension, in the topological sense, not exceeding $N$.

Détail

Publié le : 1981-01-01

MR 83d:32019 | Zbl 0439.32007 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.837

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