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

Rawat, Rama; Srivastava, R.K.

doi:10.5802/aif.2498

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 ¹ ; Srivastava, R.K.²

¹ Indian Institute of Technology Department of Mathematics and Statistics, Kanpur 208 016 (India)
² Indian Institute of Technology Department of Mathematics and Statistics Kanpur 208 016 (India)

Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2509-2523.

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.

DOI : 10.5802/aif.2498

Classification : 43A85, 44A35, 53C65
Keywords: Heisenberg group, twisted spherical means, twisted convolution, spherical harmonics, support theorems
Mot clés : groupe d’Heisenberg, moyennes sphériques tordues, convolution tordue, harmoniques sphériques, théorèmes de supports

Affiliations des auteurs :

Rawat, Rama ¹ ; Srivastava, R.K. ²

¹ Indian Institute of Technology Department of Mathematics and Statistics, Kanpur 208 016 (India)
² Indian Institute of Technology Department of Mathematics and Statistics Kanpur 208 016 (India)

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     title = {Twisted spherical means in annular regions in $\mathbb{C}^n$ and support theorems},
     journal = {Annales de l'Institut Fourier},
     pages = {2509--2523},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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     year = {2009},
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     mrnumber = {2640928},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2498/}
}

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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) no. 6, pp. 2509-2523. doi : 10.5802/aif.2498. http://www.numdam.org/articles/10.5802/aif.2498/

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