## On the extension of $h^{p}$-CR distributions defined on rough tubes

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- by G. Hoepfner, J. Hounie and L. A. Carvalho dos Santos PDF
- Proc. Amer. Math. Soc.
**140**(2012), 627-633 Request permission

## Abstract:

We consider rough tubes $X+i\mathbb R^m\subset \mathbb {C}^m$ and generalized $CR$ functions in $L^\infty (X,h^p(\mathbb R^m))$, where $h^p(\mathbb R^m)$, $0<p<\infty$, is Goldberg’s semilocal Hardy space. We show that if $X$ is arcwise connected by rectifiable arcs, 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 the authors.## References

- M. S. Baouendi and F. Trèves,
*A property of the functions and distributions annihilated by a locally integrable system of complex vector fields*, Ann. of Math. (2)**113**(1981), no. 2, 387–421. MR**607899**, DOI 10.2307/2006990 - Shiferaw Berhanu, Paulo D. Cordaro, and Jorge Hounie,
*An introduction to involutive structures*, New Mathematical Monographs, vol. 6, Cambridge University Press, Cambridge, 2008. MR**2397326**, DOI 10.1017/CBO9780511543067 - S. Berhanu and J. Hounie,
*A generalization of Bochner’s extension theorem to rough tubes,*J. Geom. Anal.**21**(2011), 455–475. - S. Bochner,
*A theorem on analytic continuation of functions in several variables*, Ann. of Math. (2)**39**(1938), no. 1, 14–19. MR**1503384**, DOI 10.2307/1968709 - Salomon Bochner and William Ted Martin,
*Several Complex Variables*, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. MR**0027863** - Albert Boggess,
*CR manifolds and the tangential Cauchy-Riemann complex*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. MR**1211412** - Al Boggess,
*The holomorphic extension of $H^p$-CR functions on tube submanifolds*, Proc. Amer. Math. Soc.**127**(1999), no. 5, 1427–1435. MR**1600104**, DOI 10.1090/S0002-9939-99-04828-5 - André Boivin and Roman Dwilewicz,
*Extension and approximation of CR functions on tube manifolds*, Trans. Amer. Math. Soc.**350**(1998), no. 5, 1945–1956. MR**1443864**, DOI 10.1090/S0002-9947-98-02019-4 - David Goldberg,
*A local version of real Hardy spaces*, Duke Math. J.**46**(1979), no. 1, 27–42. MR**523600** - G. Hoepfner, J. Hounie and L. A. Carvalho dos Santos,
*Tube structures, Hardy spaces and extension of CR distributions*, Trans. Amer. Math. Soc., to appear. - M. Kazlow,
*CR functions and tube manifolds*, Trans. Amer. Math. Soc.**255**(1979), 153–171. MR**542875**, DOI 10.1090/S0002-9947-1979-0542875-5 - Hikosaburo Komatsu,
*A local version of Bochner’s tube theorem*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**19**(1972), 201–214. MR**316749** - François Trèves,
*Hypo-analytic structures*, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. MR**1200459**

## Additional Information

**G. Hoepfner**- Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, 13565-905, Brasil
- MR Author ID: 768261
- ORCID: 0000-0002-4639-7539
- Email: hoepfner@dm.ufscar.br
**J. Hounie**- Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, 13565-905, Brasil
- MR Author ID: 88720
- 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
- Received by editor(s): September 1, 2010
- Received by editor(s) in revised form: September 7, 2010, and December 1, 2010
- Published electronically: June 17, 2011
- Additional Notes: Work supported in part by CNPq and FAPESP
- Communicated by: Franc Forstneric
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**140**(2012), 627-633 - MSC (2010): Primary 32A35, 32V25, 35N10; Secondary 42B30
- DOI: https://doi.org/10.1090/S0002-9939-2011-10927-4
- MathSciNet review: 2846332