Geodesic
Analogs of the Globevnik Problem on Riemannian Two-Point Homogeneous Spaces
Matematičeskie zametki, Tome 101 (2017) no. 3, pp. 359-372.

Voir la notice de l'article provenant de la source Math-Net.Ru

On a two-point homogeneous space $X$, we consider the problem of describing the set of continuous functions having zero integrals over all spheres enclosing the given ball. We obtain the solution of this problem and its generalizations for an annular domain in $X$. By way of applications, we prove new uniqueness theorems for functions with zero spherical means.
Keywords: spherical means, two-point homogeneous space, transmutation operator. 
@article{MZM_2017_101_3_a3,
     author = {V. V. Volchkov and Vit. V. Volchkov},
     title = {Analogs of the {Globevnik} {Problem} on {Riemannian} {Two-Point} {Homogeneous} {Spaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {359--372},
     publisher = {mathdoc},
     volume = {101},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_101_3_a3/}
}
TY  - JOUR
AU  - V. V. Volchkov
AU  - Vit. V. Volchkov
TI  - Analogs of the Globevnik Problem on Riemannian Two-Point Homogeneous Spaces
JO  - Matematičeskie zametki
PY  - 2017
SP  - 359
EP  - 372
VL  - 101
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2017_101_3_a3/
LA  - ru
ID  - MZM_2017_101_3_a3
ER  - 
%0 Journal Article
%A V. V. Volchkov
%A Vit. V. Volchkov
%T Analogs of the Globevnik Problem on Riemannian Two-Point Homogeneous Spaces
%J Matematičeskie zametki
%D 2017
%P 359-372
%V 101
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2017_101_3_a3/
%G ru
%F MZM_2017_101_3_a3
V. V. Volchkov; Vit. V. Volchkov. Analogs of the Globevnik Problem on Riemannian Two-Point Homogeneous Spaces. Matematičeskie zametki, Tome 101 (2017) no. 3, pp. 359-372. http://geodesic.mathdoc.fr/item/MZM_2017_101_3_a3/

[1] S. Helgason, “A duality in integral geometry: some generalizations of the Radon transform”, Bull. Amer. Math. Soc., 70 (1964), 435–446 | DOI | MR | Zbl

[2] S. Helgason, Integral Geometry and Radon Transforms, Springer-Verlag, New York, 2011 | MR | Zbl

[3] V. V. Volchkov, Integral Geometry and Convolution Equations, Kluwer Acad. Publ., Dordrecht, 2003 | MR | Zbl

[4] S. Helgason, “A duality in integral geometry on symmetric spaces”, Proc. U. S.–Japan Seminar in Differential Geometry, Nippon Hyoronsha, Tokyo, 1966, 37–56 | MR | Zbl

[5] E. Grinberg, E. T. Quinto, “Morera theorems for complex manifolds”, J. Funct. Anal., 178:1 (2000), 1–22 | DOI | MR | Zbl

[6] J. Globevnik, “Zero integrals on circles and characterizations of harmonic and analytic functions”, Trans. Amer. Math. Soc., 317 (1990), 313–330 | DOI | MR | Zbl

[7] C. L. Epstein, B. Kleiner, “Spherical means in annular regions”, Comm. Pure Appl. Math., 46:3 (1993), 441–451 | DOI | MR | Zbl

[8] V. V. Volchkov, “Sfericheskie srednie na evklidovykh prostranstvakh”, Ukr. matem. zhurn., 50:10 (1998), 1310–1315 | MR | Zbl

[9] V. V. Volchkov, “Sharovye srednie na simmetricheskikh prostranstvakh”, Dop. NAN Ukra{ï}ni, 2002, no. 3, 15–19 | MR | Zbl

[10] R. Rawat, R. K. Srivastava, “Spherical means in annular regions in the $n$-dimensional real hyperbolic spaces”, Proc. Indian Acad. Sci. Math. Sci., 121:3 (2011), 311–325 | DOI | MR | Zbl

[11] S. Khelgason, Gruppy i geometricheskii analiz, Mir, M., 1987 | MR

[12] V. V. Volchkov, Vit. V. Volchkov, Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group, Springer-Verlag, London, 2009 | MR | Zbl

[13] S. Helgason, Geometric Analysis on Symmetric Spaces, Math. Surveys Monogr., 39, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl

[14] S. Khelgason, Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964

[15] V. V. Volchkov, Vit. V. Volchkov, Offbeat Integral Geometry on Symmetric Spaces, Birkhäuser-Verlag, Basel, 2013 | MR | Zbl

[16] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1973 ; Г. Бейтмен, А. Эрдейи, Высшие трансцендентные функции. Функции Бесселя, функции параболического цилиндра, ортогональные многочлены, Справочная математическая библиотека, Наука, М., 1974 | MR | Zbl | MR | Zbl

[17] N. Ya. Vilenkin, Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1991 | MR | Zbl