# zbMATH — the first resource for mathematics

Maximum modulus sets in pseudoconvex boundaries. (English) Zbl 0772.32012
We quote the author’s abstract: “Let $$D$$ be a strictly pseudoconvex domain in $$\mathbb{C}^ n$$ with $$C^ \infty$$ boundary. We denote by $$A^ \infty(D)$$ the set of holomorphic functions in $$D$$ that have a $$C^ \infty$$ extension to $$\overline D$$. A closed subset $$E$$ of $$\partial D$$ is locally a maximum modulus set of $$A^ \infty(D)$$ if for every $$p\in E$$ there exists a neighborhood $$U$$ of $$p$$ and $$f\in A^ \infty(D\cap U)$$ such that $$| f|=1$$ on $$E\cap U$$ and $$| f|<1$$ on $$\overline D\cap U\backslash E$$. A submanifold $$M$$ of $$\partial D$$ is an interpolation manifold if $$T_ p(M)\subset T^ c_ p(\partial D)$$ for every $$p\in M$$, where $$T^ c_ p(\partial D)$$ is the maximal complex subspace of the tangent space $$T_ p(\partial D)$$. We prove that a local maximum modulus set for $$A^ \infty(D)$$ is locally contained in totally real $$n$$-dimensional submanifolds of $$\partial D$$ that admit a unique foliation by $$(n-1)$$-dimensional interpolation submanifolds. Let $$D=D_ 1\times\cdots\times D_ r\subset\mathbb{C}^ n$$ where $$D_ i$$ is a strictly pseudoconvex domain with $$C^ \infty$$ boundary in $$\mathbb{C}^{n_ i}$$, $$i=1,\dots,r$$. A submanifold $$M$$ of $$\partial D_ 1\times\cdots\times\partial D_ r$$ verifies the cone condition if $$\Pi_ p(T_ p(M))\cap\overline C[Jn_ 1(p),\dots,Jn_ r(p)]=\{0\}$$ for every $$p\in M$$, where $$n_ i(p)$$ is the outer normal to $$D_ i$$ at $$p$$, $$J$$ is the complex structure of $$\mathbb{C}^ n$$, $$\overline C[Jn_ 1(p),\dots,Jn_ r(p)]$$ is the closed positive cone of the real space $$V_ p$$ generated by $$Jn_ 1(p),\dots,Jn_ r(p)$$, and $$\Pi_ p$$ is the orthogonal projection of $$T_ p(\partial D)$$ on $$V_ p$$. We prove that a closed subset $$E$$ of $$\partial D_ 1\times\cdots\times\partial D_ r$$ which is locally a maximum modulus set for $$A^ \infty(D)$$ is locally contained in $$n$$-dimensional totally real submanifolds of $$\partial D_ 1\times\cdots\times\partial D_ r$$ that admit a foliation by $$(n-1)$$-dimensional submanifolds such that each leaf verifies the cone condition at every point of $$E$$. A characterization of the local peak subsets of $$\partial D_ 1\times\cdots\times\partial D_ r$$ is also given”.

##### MSC:
 32A38 Algebras of holomorphic functions of several complex variables 32T99 Pseudoconvex domains 32V40 Real submanifolds in complex manifolds
Full Text:
##### References:
  Arnold, V. I. Les Méthodes Mathématiques de la Mécanique Classique. Moscou: Editions Mir 1976.  Bedford E. $$(\partial \bar \partial )_b$$ and the real parts of CR functions. Indiana Univ. Math. J.29, 333–340 (1980). · Zbl 0441.32008 · doi:10.1512/iumj.1980.29.29024  Burns D., and Stout, E. L. Extending functions from submanifolds of the boundary. Duke Math. J.43, 391–404 (1976). · Zbl 0328.32013 · doi:10.1215/S0012-7094-76-04335-0  Chaumat J., and Chollet A.-M. Ensembles pics pourA D). Ann. Inst. Fourier29(3), 171–200 (1979). · Zbl 0398.32004 · doi:10.5802/aif.757  Chaumat, J., and Chollet, A.-M. Caractérisation et propriétés des ensembles localement pics deA D). Duke Math. J.47, 763–787 (1980). · Zbl 0454.32013 · doi:10.1215/S0012-7094-80-04745-6  Duchamp, Th., and Stout, E. L. Maximum modulus sets. Ann. Inst. Fourier31(3), 37–69 (1981). · Zbl 0439.32007 · doi:10.5802/aif.837  Fornaess, J. E., and Henriksen, B. S. Characterisation of global peak sets forA D). Math. Ann.259, 125–130 (1982). · Zbl 0489.32010 · doi:10.1007/BF01456835  Hakim, M., and Sibony, N. Ensembles pics dans des domaines strictement pseudoconexes. Duke Math. J.45, 601–617 (1978). · Zbl 0402.32008 · doi:10.1215/S0012-7094-78-04527-1  Harvey, F. R., and Wells, R. O., Jr. Holomorphic approximation and hyperfunction theory on aC 1 totally real submanifold of a complex manifold. Math. Ann.197, 287–318 (1972). · Zbl 0246.32019 · doi:10.1007/BF01428202  Henriksen, B. S. A peak set of Hausdorff dimension 2n for the algebraA(D) in the boundary of a domainD withC boundary in $$\mathbb{C}$$ n . Math. Ann.259, 271–277 (1982). · Zbl 0483.32011 · doi:10.1007/BF01457313  Hörmander, L. An Introduction to Complex Analysis in Several Variables. Princeton, NJ: Van Nostrand 1966. · Zbl 0138.06203  Iordan, A. Peak sets in weakly pseudoconvex domains. Math. Z.188, 171–185 (1985). · Zbl 0551.32016 · doi:10.1007/BF01304206  Iordan, A. Zero-sets of strictlyq-pseudoconvex functions and maximum modulus sets for CR functions. Rev. Roumaine Math. Pures et Appl.31, 303–307 (1986). · Zbl 0611.32017  Iordan, A. Ensembles de module maximal dans des domaines pseudo-convexes. C. R. Acad. Sc. Paris, t. 300 serie I, no. 19, 655–656 (1985). · Zbl 0585.32021  Jimbo, T., and Sakai, A. Interpolation manifolds for products of strictly pseudoconvex domains. Complex Variables8, 333–341 (1987). · Zbl 0587.32032  Labonde, J. M. Thesis, Université Paris-Sud, 1985.  Løw, E. Inner functions and boundaries values inH $$\Omega$$) andA($$\Omega$$) in smoothly bounded pseudoconvex domains. Math. Z.185, 191–210 (1984). · Zbl 0526.32017 · doi:10.1007/BF01181690  Noell, A. V. Properties of peak sets in weakly pseudoconvex boundaries inC 2. Math. Z.185, 99–116 (1984). · Zbl 0531.32011 · doi:10.1007/BF01215494  Rosay, J.-P. Personal communication.  Saerens, R. Interpolation manifolds. Ann. Sc. Norm. Sup. Pisa, C. Sci.11, 177–211 (1984). · Zbl 0579.32023  Saerens, R. Interpolation theory in $$\mathbb{C}$$ n : A survey. In: Complex Analysis. Lecture Notes in Mathematics, Vol. 1268, pp. 158–188. New York: Springer-Verlag 1987. · Zbl 0646.32011  Suerens, R., and Stout, E. L. Differentiable interpolation on the polydisc. Complex Variables2, 271–282 (1984). · Zbl 0586.32020  Sakai, A. Maximum modulus sets forA D). Kobe J. Math.2, 85–87 (1985). · Zbl 0583.32042  Sibony, N. Valeurs au bord de fonctions holomorphes et ensembles polynomialement convexes. In: Séminaire Pierre Lelong, 1975–1976, Springer Lecture Notes in Mathematics, Vol. 578, pp. 300–313. Berlin: Springer-Verlag 1977. · Zbl 0382.32004  Stout, E. L. Interpolation manifolds. In: Recent Developments in Several Complex Variables, Annals of Mathematics Studies, pp. 373–391, Princeton, NJ: Princeton University Press 1981.  Stout, E. L. The dimension of peak-interpolation sets. Proc. Am. Math. Soc.86, 413–416 (1982). · Zbl 0502.32012 · doi:10.1090/S0002-9939-1982-0671206-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.