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

We investigate some aspects of maximum modulus sets in the boundary of a strictly pseudoconvex domain D of dimension N. If ΣbD 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 fA(D) has dimension, in the topological sense, not exceeding N.

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 ΣbD est une variété différentiable de dimension réelle N, et si Σ est l’ensemble des points où la valeur absolue d’une fonction fA 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 fA(D), la dimension topologique de l’ensemble des points où |f| prend son maximum est au plus N.

@article{AIF_1981__31_3_37_0,
     author = {Duchamp, Thomas and Stout, Edgar Lee},
     title = {Maximum modulus sets},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {31},
     number = {3},
     year = {1981},
     pages = {37-69},
     doi = {10.5802/aif.837},
     zbl = {0439.32007},
     mrnumber = {83d:32019},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1981__31_3_37_0}
}
Duchamp, Thomas; Stout, Edgar Lee. Maximum modulus sets. Annales de l'Institut Fourier, Volume 31 (1981) no. 3, pp. 37-69. doi : 10.5802/aif.837. http://www.numdam.org/item/AIF_1981__31_3_37_0/

[1] H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in Cn, Amer. J. Math., 93 (1971), 65-74. | MR 44 #1841 | Zbl 0221.32011

[2] A. Andreotti and R. Narasimhan, A topological property of Runge pairs, Ann. Math., (2) 76 (1962), 499-509. | MR 25 #4128 | Zbl 0178.42703

[3] E. Bishop, A generalization of the Stone-Weierstrass theorem, Pacific J. Math., 11 (1961), 777-783. | MR 24 #A3502 | Zbl 0104.09002

[4] D.E. Blair, Contact Manifolds in Riemannian Geometry, Springer Lecture Notes in Mathematics, vol. 509, Springer-Verlag, Berlin, Heidelberg, New York, 1976. | MR 57 #7444 | Zbl 0319.53026

[5] A. Browder, Cohomology of maximal ideal spaces, Bull. Amer. Math. Soc., 67 (1961), 515-516. | MR 24 #A440 | Zbl 0107.09501

[6] D. Burns and E.L. Stout, Extending functions from submanifolds of the boundary, Duke Math., J., 43 (1976), 391-404. | MR 54 #3028 | Zbl 0328.32013

[7] H. Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France, 85 (1957), 77-99. | Numdam | MR 20 #1339 | Zbl 0083.30502

[8] J. Chaumat and A.M. Chollet, Ensembles pics pour A∞ (D), Ann. Inst. Fourier, Grenoble, XXIX (1979), 171-200. | Numdam | MR 81c:32036 | Zbl 0398.32004

[9] A.M. Davie and B. Øksendal, Peak interpolation sets for some algebras of analytic functions, Pacific J. Math., 41 (1972), 81-87. | MR 46 #9394 | Zbl 0232.46055

[10] H. Federer, Geometric Measure Theory, Springer-Verlag New York, Inc., New York, 1969. | MR 41 #1976 | Zbl 0176.00801

[11] T. Duchamp, The classification of Legendre embeddings, to appear.

[12] J.E. Fornaess, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math., 98 (1976), 529-569. | MR 54 #10669 | Zbl 0334.32020

[13] R. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. | MR 31 #4927 | Zbl 0141.08601

[14] C.D. Hill and G. Taiani, Families of analytic discs in Cn with boundaries on a prescribed CR submanifold, Ann. Scuola Norm. Sup. Pisa Sci., (IV) V, (1978), 327-380. | Numdam | MR 80c:32023 | Zbl 0399.32008

[15] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, 1962. | MR 24 #A2844 | Zbl 0117.34001

[16] H. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton, 1948. | Zbl 0036.12501

[17] V.S. Klein, Behavior of Holomorphic Functions at Generating Submanifolds of the Boundary, doctoral dissertation, University of Washington, Seattle, 1979.

[18] H.B. Lawson, Lectures on the Quantitative Theory of Foliations, CBMS Regional Conference Series in Mathematics, Number 27, American Mathematical Society, Providence, Rhode Island, 1977. | Zbl 0343.57014

[19] L. Loomis and S. Sternberg, Advanced Calculus, Addison-Wesley, Reading, 1968. | MR 37 #2912 | Zbl 0162.35301

[20] J. Milnor, Topology from the Differentiable Viewpoint, University Press of Virginia, Charlottesville, 1965. | MR 37 #2239 | Zbl 0136.20402

[21] M. Müller, Geometrisch Untersuchungen allgemeiner und einiger spezieller Pseudokonvexer Gebiete, Bonner Math. Schriften, 78, Bonn, 1975. | Zbl 0331.32017

[22] S.I. Pinchuk, A boundary uniqueness theorem for holomorphic functions of several complex variables, Math. Notes, 15 (1974), 116-120. | MR 50 #2558 | Zbl 0292.32002

[23] M. Range and Y.-T. Siu, Ck approximation by holomorphic functions and ATT-closed forms on Ck submanifolds of a complex manifold, Math. Ann., 210 (1974), 105-122. | Zbl 0275.32008

[24] G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, Act. Sci. Indust., 1183, Hermann, Paris, 1952. | MR 14,1113a | Zbl 0049.12602

[25] W. Rudin, Peak interpolation manifolds of class C1, Pacific J. Math., 75 (1978), 267-279. | MR 58 #6346 | Zbl 0383.32007

[26] W. Rudin, Lectures on the Edge-of-the-Wedge Theorem, CBMS Regional Conference Series in Mathematics, Number 6, American Mathematical Society, Providence, Rhode Island, 1971. | MR 46 #9389 | Zbl 0214.09001

[27] W. Rudin and E.L. Stout, Boundary properties of functions of several complex variables, J. Math. Mech., 14 (1965), 991-1006. | MR 32 #230 | Zbl 0147.11601

[28] A. Sadullaev, A boundary uniqueness theorem in Cn, Math. USSR Sbornik, 30 (1976), 501-514. | Zbl 0385.32007

[29] J. Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969. | MR 55 #6457 | Zbl 0203.14501

[30] B. Shiffman, On the continuation of analytic curves, Math. Ann., 184 (1970), 268-274. | MR 46 #3828 | Zbl 0176.38003

[31] N. Sibony, Valeurs au bord de fonctions holomorphes et ensembles polynomialement convexes, Séminaire Pierre Lelong 1975-1976. Springer Lecture Notes in Mathematics, vol. 578, Springer-Verlag, Berlin, Heidelberg, New York, 1977. | Zbl 0382.32004

[32] K. Stein, Analytische Projektion komplexer Mannigfaltigkeiten, Colloque sur les Fonctions de Plusieurs Variables, Brussels, 1953. George Throne, Leige and Masson, Paris, 1953. | Zbl 0052.08604

[32a] K. Stein, Die Existenz Komplexer Basen zu holomorphen Abbildungen, Math. Ann., 136 (1958), 1-8. | MR 20 #4657 | Zbl 0081.30202

[33] S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, 1964. | MR 33 #1797 | Zbl 0129.13102

[34] E.L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, Tarrytown-on-Hudson and Belmont, 1971. | MR 54 #11066 | Zbl 0286.46049

[35] E.L. Stout, Interpolation manifolds, Recent Developments in Several Complex Variables, Annals of Mathematics Studies, to appear. | Zbl 0486.32010

[36] A.E. Tumanov, A peak set for the disc algebra of metric dimension 2.5 in the three-dimensional unit sphere, Math. USSR Izvestija, 11 (1977), 370-377. | MR 58 #6349 | Zbl 0379.46048

[37] B.M. Weinstock, Zero-sets of continuous holomorphic functions on the boundary of a strongly pseudoconvex domain, J. London Math. Soc., 18 (1978), 484-488. | MR 80e:32010 | Zbl 0413.32008

[38] R.O. Wells, Compact real submanifolds of a complex manifold with nondegenerate holomorphic tangent bundles, Math. Ann., 179 (1969), 123-129. | MR 38 #6104 | Zbl 0167.21604

[39] R.O. Wells, Real analytic subvarieties and holomorphic approximation, Math. Ann., 179 (1969), 130-141. | MR 39 #476 | Zbl 0167.06704

[40] A. Zygmund, Trigonometric Series, vol. I., Cambridge University Press, Cambridge, 1959. | Zbl 0085.05601