Behaviour at infinity of solutions of twisted convolution equations
Izvestiya. Mathematics , Tome 76 (2012) no. 1, pp. 79-93.

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We obtain a precise characterization of the minimal rate of growth at infinity of non-trivial solutions of twisted convolution equations in unbounded domains of $\mathbb{C}^n$. As an application, we obtain definitive versions of the two-radii theorem for twisted spherical means.
Keywords: Heisenberg group, twisted convolution equation, two-radii theorem, confluent hypergeometric functions.
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V. V. Volchkov; Vit. V. Volchkov. Behaviour at infinity of solutions of twisted convolution equations. Izvestiya. Mathematics , Tome 76 (2012) no. 1, pp. 79-93. http://geodesic.mathdoc.fr/item/IM2_2012_76_1_a2/

[1] G. B. Folland, Harmonic analysis in phase space, Ann. of Math. Stud., 122, Princeton Univ. Press, Princeton, NJ, 1989 | MR | Zbl

[2] S. Thangavelu, Lectures on Hermite and Laguerre expansions, Math. Notes, 42, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl

[3] C. Berenstein, D.-Ch. Chang, J. Tie, Laguerre calculus and its applications on the Heisenberg group, AMS/IP Stud. Adv. Math., 22, Amer. Math. Soc., Providence, RI, 2001 | MR | Zbl

[4] F. John, Plane waves and spherical means applied to partial differential equations, Interscience Publ., New York–London, 1955 | MR | Zbl

[5] J. D. Smith, “Harmonic analysis of scalar and vector fields in $\mathbb{R}^n$”, Proc. Cambridge Philos. Soc., 72:3 (1972), 403–416 | DOI | MR | Zbl

[6] A. Sitaram, “Fourier analysis and determining sets for Radon measures on $\mathbb{R}^n$”, Illinois J. Math., 28:2 (1984), 339–347 | MR | Zbl

[7] S. Thangavelu, “Spherical means and $\mathrm{CR}$ functions on the Heisenberg group”, J. Anal. Math., 63:1 (1994), 255–286 | DOI | MR | Zbl

[8] R. Rawat, A. Sitaram, “Injectivity sets for spherical means on $\mathbb{R}^n$ and on symmetric spaces”, J. Fourier Anal. Appl., 6:3 (2000), 343–348 | DOI | MR | Zbl

[9] L. Flatto, “The converse of Gauss's theorem for harmonic functions”, J. Differential Equations, 1:4 (1965), 483–490 | DOI | MR | Zbl

[10] O. A. Ochakovskaya, “On functions with zero integrals over balls of fixed radius on a half-space”, Dokl. Math., 64:3 (2001), 413–415 | MR | Zbl

[11] O. A. Ochakovskaya, “Liouville-type theorems for functions with zero integrals over balls of fixed radius”, Dokl. Math., 76:1 (2007), 530–532 | DOI | MR | Zbl

[12] O. A. Ochakovskaya, “Precise characterizations of admissible rate of decrease of a non-trivial function with zero ball means”, Sb. Math., 199:1 (2008), 45–65 | DOI | MR | Zbl

[13] O. A. Ochakovskaya, “Majorants of functions with zero integrals over balls of fixed radius”, Dokl. Math., 77:3 (2008), 446–448 | DOI | MR | Zbl

[14] V. V. Volchkov, Integral geometry and convolution equations, Kluwer Acad. Publ., Dordrecht, 2003 | MR | Zbl

[15] S. Thangavelu, “Mean periodic functions on phase space and the Pompeiu problem with a twist”, Ann. Inst. Fourier (Grenoble), 45:4 (1995), 1007–1035 | DOI | MR | Zbl

[16] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, vols. I, II, McGraw-Hill, New York–Toronto–London, 1953 | MR | MR | MR | Zbl | Zbl

[17] V. V. Volchkov, V. V. Volchkov, “Convolution equations in many-dimensional domains and on the Heisenberg reduced group”, Sb. Math., 199:8 (2008), 1139–1168 | DOI | MR

[18] M. Shahshahani, A. Sitaram, “The Pompeiu problem in exterior domains in symmetric spaces”, Integral geometry (Brunswick, Maine, 1984), Contemp. Math., 63, Amer. Math. Soc., Providence, RI, 1987, 267–277 | MR | Zbl

[19] V. V. Volchkov, “Theorems on ball mean values in symmetric spaces”, Sb. Math., 192:9 (2001), 1275–1296 | DOI | MR | Zbl

[20] M. A. Evgrafov, Asymptotic estimates and entire functions, Gordon and Breach, New York, 1961 | MR | MR | Zbl | Zbl

[21] W. Rudin, Function theory in the unit ball of $\mathbb C^n$, Grundlehren Math. Wiss., 241, Springer-Verlag, New York–Heidelberg–Berlin, 1980 | MR | MR | Zbl | Zbl