Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 

 

Plurisubharmonic domination


Author: László Lempert
Journal: J. Amer. Math. Soc. 17 (2004), 361-372
MSC (2000): Primary 32Txx, 32U05, 46G20
Published electronically: November 25, 2003
MathSciNet review: 2051614
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Abstract: For a large class of separable Banach spaces $X$ we prove the following. Given a pseudoconvex open $\Omega \subset X$ and $u:\Omega\to\mathbb{R}$ that is locally bounded above, there is a plurisubharmonic $v:\Omega\to\mathbb{R}$ such that $u\le v$. We also discuss applications of this result.


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

  • [A] A. Arroud, Plongement des variétés analytiques complexes de dimension infinie, Thesis, Lille (1983).
  • [B] Errett Bishop, Mappings of partially analytic spaces, Amer. J. Math. 83 (1961), 209–242. MR 0123732
  • [BF] Robert Bonic and John Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877–898. MR 0198492
  • [D] Seán Dineen, Bounding subsets of a Banach space, Math. Ann. 192 (1971), 61–70. MR 0303290
  • [H] Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
  • [J1] Bengt Josefson, Bounding subsets of 𝑙^{∞}(𝐴), J. Math. Pures Appl. (9) 57 (1978), no. 4, 397–421. MR 524627
  • [J2] B. Josefson, Approximations of holomorphic functions in certain Banach spaces, manuscript (2000).
  • [L1] László Lempert, The Dolbeault complex in infinite dimensions. I, J. Amer. Math. Soc. 11 (1998), no. 3, 485–520. MR 1603858, 10.1090/S0894-0347-98-00266-5
  • [L2] László Lempert, Approximation of holomorphic functions of infinitely many variables. II, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 423–442 (English, with English and French summaries). MR 1775356
  • [L3] László Lempert, The Dolbeault complex in infinite dimensions. III. Sheaf cohomology in Banach spaces, Invent. Math. 142 (2000), no. 3, 579–603. MR 1804162, 10.1007/PL00005794
  • [L4] L. Lempert, Vanishing cohomology for holomorphic vector bundles in a Banach setting, Asian J. Math. to appear.
  • [Ma] P. Mazet, Letter of July 1998.
  • [Mu] Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de Matemática [Mathematical Notes], 107. MR 842435
  • [Na] Raghavan Narasimhan, Imbedding of holomorphically complete complex spaces, Amer. J. Math. 82 (1960), 917–934. MR 0148942
  • [No] Philippe Noverraz, Pseudo-convexité, convexité polynomiale et domaines d’holomorphie en dimension infinie, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 3. Notas de Matemática (48). MR 0358348
  • [P] Imre Patyi, On the \overline∂-equation in a Banach space, Bull. Soc. Math. France 128 (2000), no. 3, 391–406 (English, with English and French summaries). MR 1792475
  • [R] R. Remmert, Habilitationsschrift, Münster (1958).
  • [Sn] Ivan Singer, Bases in Banach spaces. I, Springer-Verlag, New York-Berlin, 1970. Die Grundlehren der mathematischen Wissenschaften, Band 154. MR 0298399
    Ivan Singer, Bases in Banach spaces. II, Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. MR 610799
  • [Su] Yum Tong Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976/77), no. 1, 89–100. MR 0435447

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Additional Information

László Lempert
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: lempert@math.purdue.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-03-00448-X
Received by editor(s): March 6, 2003
Published electronically: November 25, 2003
Additional Notes: Research partially supported by an NSF grant
Article copyright: © Copyright 2003 American Mathematical Society