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Transactions of the American Mathematical Society

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Tube structures, Hardy spaces and extension of CR distributions


Authors: G. Hoepfner, J. Hounie and L. A. Carvalho dos Santos
Journal: Trans. Amer. Math. Soc. 363 (2011), 5091-5109
MSC (2000): Primary 32A35, 32V25, 35N10; Secondary 42B30
DOI: https://doi.org/10.1090/S0002-9947-2011-05138-X
Published electronically: May 11, 2011
MathSciNet review: 2813409
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Abstract: We consider rough tubes $ X+i\mathbb{R}^m\subset\mathbb{C}^m$, where $ X\subset \mathbb{R}^m$ is a measurable set, and extend the notion of $ CR$ function to the space $ L^\infty(X,h^p(\mathbb{R}^m))$, where $ h^p(\mathbb{R}^m)$, $ 0<p<\infty$, is Goldberg's semilocal Hardy space. We show that if $ X$ is the image of some connected manifold by some $ C^1$ map, then all such $ CR$ functions can be extended to the convex hull of the tube as $ CR$ functions $ \in L^\infty(\mathrm{ch}(X),h^p(\mathbb{R}^m))$. This extends previous work of Boggess.


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

  • [BT] M. S. Baouendi and F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. (2) 113 (1981), 387-421. MR 607899 (82f:35057)
  • [BCH] S. Berhanu, P. Cordaro and J. Hounie, An Introduction to Involutive Structures, Cambridge University Press, 2008. MR 2397326 (2009b:32048)
  • [Bo] S. Bochner, A theorem on analytic continuation of functions in several variables Ann. of Math. (2) 39 (1938), 14-19. MR 1503384
  • [BM] S. Bochner and W. T. Martin, Functions of several complex variables, Princeton University Press, 1948. MR 0027863 (10:366a)
  • [B1] A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics (1991), CRC Press. MR 1211412 (94e:32035)
  • [B2] A. Boggess, The holomorphic extension of $ H^p$-CR functions on tube submanifolds, Proc. Amer. Math. Soc. 127 (1999), 1427-1435. MR 1600104 (99h:32012)
  • [BD] Boivin and Dwilewicz, Extension and Approximation of CR Functions on Tube Manifolds, Trans. Amer. Math. Soc. 350 (1998), 1945-1956. MR 1443864 (98h:32011)
  • [Du] P. Duren, Theory of $ H^p$ spaces, Academic Press (1970). MR 0268655 (42:3552)
  • [G] D. Goldberg, A local version of real Hardy spaces, Duke Math. J., 46 (1979), 27-42. MR 523600 (80h:46052)
  • [HM] J. Hounie and P. Malagutti, On the convergence of the Baouendi-Treves approximation formula, Comm. P.D.E., 23 (1998), 1305-1347. MR 1642611 (2000b:32072)
  • [H] L. Hörmander, Notions of convexity, Birkhäuser (1994). MR 1301332 (95k:00002)

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Additional Information

G. Hoepfner
Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, 13565-905, Brasil
Email: hoepfner@dm.ufscar.br

J. Hounie
Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, 13565-905, Brasil
Email: hounie@dm.ufscar.br

L. A. Carvalho dos Santos
Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, 13565-905, Brasil
Email: luis@dm.ufscar.br

DOI: https://doi.org/10.1090/S0002-9947-2011-05138-X
Keywords: Locally integrable structures, Baouendi-Treves approximation formula, Hardy spaces, CR functions
Received by editor(s): May 11, 2009
Published electronically: May 11, 2011
Additional Notes: This work was supported in part by CNPq and FAPESP
Article copyright: © Copyright 2011 American Mathematical Society

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