Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems

Rawat, Rama; Srivastava, R.K.

Twisted spherical means in annular regions in

ℂ^{n}

and support theorems
[Moyennes sphériques tordues dans des anneaux de

ℂ^{n}

et théorèmes de support]

Rawat, Rama ; Srivastava, R.K.

Annales de l'Institut Fourier, Tome 59 (2009), p. 2509-2523 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

Soit $Z (Ann (r, R))$ la classe de toutes les fonctions continues sur l’anneau $Ann (r, R)$ de $ℂ^{n}$ de moyenne sphérique tordue $f \times μ_{s} (z) = 0$ , pour tout $z \in ℂ^{n}$ et $s > 0$ tels que la sphère $S_{s} (z) \subseteq Ann (r, R)$ et la boule $B_{r} (0) \subseteq B_{s} (z)$ . Dans cet article, nous donnons une caractérisation des fonctions dans $Z (Ann (r, R))$ en termes de leur coefficients dans le développement en harmoniques sphériques. Nous prouvons également des théorèmes de support pour les moyennes sphériques tordues dans $ℂ^{n}$ qui améliorent certains résultats antérieurs.

Let $Z (Ann (r, R))$ be the class of all continuous functions $f$ on the annulus $Ann (r, R)$ in $ℂ^{n}$ with twisted spherical mean $f \times μ_{s} (z) = 0,$ whenever $z \in ℂ^{n}$ and $s > 0$ satisfy the condition that the sphere $S_{s} (z) \subseteq Ann (r, R)$ and ball $B_{r} (0) \subseteq B_{s} (z) .$ In this paper, we give a characterization for functions in $Z (Ann (r, R))$ in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in $ℂ^{n}$ which improve some of the earlier results.

Publié le : 2009-01-01

MR 2640928 | Zbl pre05673904

DOI : https://doi.org/10.5802/aif.2498
Classification: 43A85, 44A35, 53C65
Mots clés: groupe d’Heisenberg, moyennes sphériques tordues, convolution tordue, harmoniques sphériques, théorèmes de supports

@article{AIF_2009__59_6_2509_0,
     author = {Rawat, Rama and Srivastava, R.K.},
     title = {Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems},
     journal = {Annales de l'Institut Fourier},
     volume = {59},
     year = {2009},
     pages = {2509-2523},
     doi = {10.5802/aif.2498},
     zbl = {pre05673904},
     mrnumber = {2640928},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2009__59_6_2509_0}
}

Rawat, Rama; Srivastava, R.K. Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems. Annales de l'Institut Fourier, Tome 59 (2009) pp. 2509-2523. doi : 10.5802/aif.2498. http://gdmltest.u-ga.fr/item/AIF_2009__59_6_2509_0/

Bibliographie

[1] Agranovsky, M. L.; Rawat, Rama Injectivity sets for spherical means on the Heisenberg group, J. Fourier Anal. Appl., Tome 5 (1999) no. 4, pp. 363-372 | Article | MR 1700090 | Zbl 0931.43007

[2] Epstein, C. L.; Kleiner, B. Spherical means in annular regions, Comm. Pure Appl. Math., Tome 46 (1993) no. 3, pp. 441-451 | Article | MR 1202964 | Zbl 0841.31006

[3] Helgason, S. The Radon Transform, Birkhauser (1983) | Zbl 0547.43001

[4] Narayanan, E. K.; Thangavelu, S. Injectivity sets for spherical means on the Heisenberg group, J. Math. Anal. Appl., Tome 263 (2001) no. 2, pp. 565-579 | Article | MR 1866065 | Zbl 0995.43003

[5] Narayanan, E. K.; Thangavelu, S. A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on $ℂ^{n}$ , Ann. Inst. Fourier, Grenoble, Tome 56 (2006) no. 2, pp. 459-473 | Article | Numdam | MR 2226023 | Zbl 1089.43006

[6] Rudin, W. Function theory in the unit ball of $ℂ^{n}$ , Springer-Verlag, New York-Berlin (1980) | MR 601594 | Zbl 0495.32001

[7] Thangavelu, S. An introduction to the uncertainty principle, Prog. Math., Birkhauser, Boston, Tome 217 (2004) | MR 2008480 | Zbl pre02039097