Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On peak-interpolation manifolds for $\boldsymbol{A}\boldsymbol{(}\boldsymbol{\Omega}\boldsymbol{)}$ for convex domains in $\boldsymbol{\mathbb{C} }^{\boldsymbol{n}}$


Author: Gautam Bharali
Journal: Trans. Amer. Math. Soc. 356 (2004), 4811-4827
MSC (2000): Primary 32A38, 32T25; Secondary 32C25, 32D99
DOI: https://doi.org/10.1090/S0002-9947-04-03705-5
Published electronically: June 22, 2004
MathSciNet review: 2084399
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\Omega$ be a bounded, weakly convex domain in ${\mathbb{C} }^n$, $n\geq 2$, having real-analytic boundary. $A(\Omega)$ is the algebra of all functions holomorphic in $\Omega$ and continuous up to the boundary. A submanifold $\boldsymbol{M}\subset \partial \Omega$ is said to be complex-tangential if $T_p(\boldsymbol{M})$ lies in the maximal complex subspace of $T_p(\partial \Omega)$ for each $p\in\boldsymbol{M}$. We show that for real-analytic submanifolds $\boldsymbol{M}\subset\partial \Omega$, if $\boldsymbol{M}$ is complex-tangential, then every compact subset of $\boldsymbol{M}$ is a peak-interpolation set for $A(\Omega)$.


References [Enhancements On Off] (What's this?)

  • 1. E. Bishop, A general Rudin-Carleson theorem, Proc. Amer. Math. Soc. 13 (1962), 140-143. MR 0133462 (24:A3293)
  • 2. T. Bloom and I. Graham, On type conditions for generic real submanifolds in $\mathbb{C} ^n$, Invent. Math. 40 (1984), 217-43. MR 0589930 (58:28644)
  • 3. D. Catlin, Boundary invariants of pseudoconvex domains, Ann. of Math.(2) 120 (1984), 529-586. MR 0769163 (86c:32019)
  • 4. J.P. D'Angelo, Real hypersurfaces, orders of contact, and applications, Annals of Math. 115 (1982), 615-637. MR 0657241 (84a:32027)
  • 5. G.M. Henkin and A.E. Tumanov, Interpolation submanifolds of pseudoconvex manifolds, Translations Amer. Math. Soc. 115 (1980), 59-69. MR 0604784 (81m:00006)
  • 6. S. \Lojasiewicz, Sur le problème de la division, Studia Math. 18 (1959), 87-136. MR 0107168 (21:5893)
  • 7. J.D. McNeal, Convex domains of finite type, J. Funct. Anal. 108 (1992), 361-373. MR 1176680 (93h:32020)
  • 8. A. Nagel, Smooth zero sets and interpolation sets for some algebras of holomorphic functions on strictly pseudoconvex domains, Duke Math. J. 43 (1976), 323-348. MR 0442284 (56:670)
  • 9. A. Nagel and W. Rudin, Local boundary behavior of bounded holomorphic functions, Canad. J. Math. 30 (1978), 583-592. MR 0486595 (58:6315)
  • 10. W. Rudin, Peak-interpolation sets of class $\mathcal{C}^1$, Pacific J. Math. 75 (1978), 267-279. MR 0486630 (58:6346)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32A38, 32T25, 32C25, 32D99

Retrieve articles in all journals with MSC (2000): 32A38, 32T25, 32C25, 32D99


Additional Information

Gautam Bharali
Affiliation: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: bharali@math.wisc.edu, bharali@umich.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03705-5
Keywords: Complex-tangential, finite type domain, interpolation set, pseudoconvex domain
Received by editor(s): July 23, 2002
Published electronically: June 22, 2004
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society