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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Carleman approximation of maps into Oka manifolds
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by Brett Chenoweth PDF
Proc. Amer. Math. Soc. 147 (2019), 4847-4861 Request permission

Abstract:

In this paper we obtain a Carleman approximation theorem for maps from Stein manifolds to Oka manifolds. More precisely, we show that under suitable complex analytic conditions on a totally real set $M$ of a Stein manifold $X$, every smooth map $X \rightarrow Y$ to an Oka manifold $Y$ satisfying the Cauchy-Riemann equations along $M$ up to order $k$ can be $\mathscr {C}^k$-Carleman approximated by holomorphic maps $X \rightarrow Y$. Moreover, if $K$ is a compact $\mathscr {O}(X)$-convex set such that $K \cup M$ is $\mathscr {O}(X)$-convex, then we can $\mathscr {C}^k$-Carleman approximate maps which satisfy the Cauchy-Riemann equations up to order $k$ along $M$ and are holomorphic on a neighbourhood of $K$ or merely in the interior of $K$ if the latter set is the closure of a strongly pseudoconvex domain.
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Additional Information
  • Brett Chenoweth
  • Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia
  • Email: brett.s.chenoweth@gmail.com
  • Received by editor(s): August 6, 2018
  • Received by editor(s) in revised form: February 13, 2019
  • Published electronically: May 29, 2019
  • Additional Notes: The author was supported by grant MR-39237 from ARRS, Republic of Slovenia, associated to the research program P1-0291 Analysis and Geometry.
  • Communicated by: Filippo Bracci
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4847-4861
  • MSC (2010): Primary 32E30, 32V40, 32E10
  • DOI: https://doi.org/10.1090/proc/14595
  • MathSciNet review: 4011518