Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

On the uniqueness question for Hahn-Banach extensions from the space of $ \mathcal{L}^1$ analytic functions


Author: Edgar Reich
Journal: Proc. Amer. Math. Soc. 88 (1983), 305-310
MSC: Primary 46E15; Secondary 30H05
DOI: https://doi.org/10.1090/S0002-9939-1983-0695263-1
MathSciNet review: 695263
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Omega $ be a region of the complex plane, $ \mathcal{B}\left( \Omega \right)$ the space of analytic $ {\mathcal{L}^1}$ functions over $ \Omega $, and $ \kappa \in {\mathcal{L}^\infty }\left( \Omega \right)$. An evident necessary condition for the linear functional $ {\Lambda _\kappa }\left[ \varphi \right] = \smallint {\smallint _\Omega }\kappa \varphi dxdy(\varphi \in \mathcal{B}(\Omega ))$ to have a unique Hahn-Banach extension from $ \mathcal{B}(\Omega )$ to $ {\mathcal{L}^1}(\Omega )$ is that $ \vert\vert {{\Lambda _{\kappa \vert G}}} \vert\vert = {\vert\vert {\kappa \vert G} \vert\vert _\infty }$ for every restriction $ \kappa \vert G$ of $ \kappa $ to a subregion $ G$ of $ \Omega $. An example is constructed to show that not even a considerably stronger condition is sufficient for uniqueness of the Hahn-Banach extension. Remarks on the problem of whether $ \left\vert {\kappa (z)} \right\vert$ is necessarily constant a.e. if the Hahn-Banach extension is unique indicate that this question is still open, contrary to an assertion in the literature.


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

  • [1] Lennart Carleson, Mergelyan's Theorem on uniform polynomial approximation, Math. Scand. 15 (1964), 167-175. MR 0198209 (33:6368)
  • [2] W. K. Hayman and Edgar Reich, On Teichmüller mappings of the disk, Complex Variables 1 (1982) (to appear). MR 674357 (84e:30031)
  • [3] Edgar Reich, Uniqueness of Hahn-Banach extensions from certain spaces of analytic functions, Math. Z. 167 (1979), 81-89. MR 532887 (80j:30074)
  • [4] -, A criterion for unique extremality of Teichmüller mappings, Indiana Univ. Math. J. 30 (1981), 441-447. MR 611232 (84e:30032)
  • [5] -, On criteria for unique extremality of Teichmüller mappings, Ann. Acad. Sci. Fenn. A.I. Math. 6 (1981), 289-302. MR 658931 (83j:30022)
  • [6] V. G. Šeretov, Locally extremal quasiconformal mappings, Dokl. Akad. Nauk SSSR 250 (1980), 1338-1340 = Soviet Math. Dokl. 21 (1980), 343-345. MR 564341 (81e:30030)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E15, 30H05

Retrieve articles in all journals with MSC: 46E15, 30H05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0695263-1
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society