no. 3
Maximum modulus sets
Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 37-69.

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 Σ⊂bD est une variété différentiable de dimension réelle N, et si Σ est l’ensemble des points où la valeur absolue d’une fonction f∈A 2 (D) atteint son maximum, alors Σ 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,R) s’annule ou si D ¯⊂C N est polynomialement convexe. Finalement, pour toute fonction f∈A(D), la dimension topologique de l’ensemble des points où |f| prend son maximum est au plus N.

We investigate some aspects of maximum modulus sets in the boundary of a strictly pseudoconvex domain D of dimension N. If Σ⊂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,R) vanishes or if D ¯⊂C N is polynomially convex. Finally, the maximum modulus set of an arbitrary f∈A(D) has dimension, in the topological sense, not exceeding N.

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     title = {Maximum modulus sets},
     journal = {Annales de l'Institut Fourier},
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Duchamp, Thomas; Stout, Edgar Lee. Maximum modulus sets. Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 37-69. doi : 10.5802/aif.837. https://aif.centre-mersenne.org/articles/10.5802/aif.837/

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