Maximum modulus sets in pseudoconvex boundaries |
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Authors: | Andrei Iordan |
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Affiliation: | 1. Analyse Complexe et Géométrie, Université Paris VI, 75252, Paris Cedex 05, France
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Abstract: | LedD be a strictly pseudoconvex domain in ? n withC ∞ boundary. We denote byA ∞(D) the set of holomorphic functions inD that have aC ∞ extension to (bar D) . A closed subsetE of ?D is locally a maximum modulus set forA ∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ∞(D∩U) such that |f|=1 onE∩U and |f|<1 on (bar D cap Ubackslash E) . A submanifoldM of ?D is an interpolation manifold ifT p (M)?T p c (?D) for everyp∈M, whereT p c (?D) is the maximal complex subspace of the tangent spaceT p (?D). We prove that a local maximum modulus set forA ∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D 1 x ... xD r ? ? n whereD i is a strictly pseudoconvex domain withC ∞ boundary in ? n i ,i=1,…,r. A submanifoldM of ?D 1×…×?D r verifies the cone condition if (II_p (T_p (M)) cap bar C[Jn_1 (p),...,Jn_r (p)] = { 0} ) for everyp∈M, wheren i (p) is the outer normal toD i atp, J is the complex structure of ? n , (bar C[Jn_1 (p),...,Jn_r (p)]) is the closed positive cone of the real spaceV p generated byJ n 1(p),…,J n r(p), and II p is the orthogonal projection ofT p (?D) onV p . We prove that a closed subsetE of ?D 1×…×?D r which is locally a maximum modulus set forA ∞(D) is locally contained inn-dimensional totally real submanifolds of ?D 1×…×?D r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D 1×…×?D r is also given. |
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