Functions with ball mean values equal to zero on compact two-point homogeneous spaces

Vit. V. Volchkov

Sbornik. Mathematics, Tome 198 (2007) no. 4, pp. 465-490

Résumé

The problem of the description of the class of functions with zero integral over each ball of prescribed radius lying in a fixed ball is solved for compact two-point homogeneous spaces. Bibliography: 35 titles.

Bibliographie

[1] K. A. Berenstein, D. Struppa, “Kompleksnyi analiz i uravneniya v svertkakh”, Itogi nauki i tekhn. Sovr. probl. matem. Fundament. napravleniya, 54, VINITI, M., 1989, 5–111 ; C. A. Berenstein, D. C. Struppa, “Complex analysis and convolution equations”, Several complex variables. V: Complex analysis in partial differential equations and mathematical physics, Encyclopedia Math. Sci., 54, Springer-Verlag, Berlin, 1993, 1–108 | MR | Zbl

[2] L. Zalcman, “A bibliographic survey of the Pompeiu problem”, Approximation by solutions of partial differential equations, Proc. NATO Adv. Res. Workshop (Hanstholm, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 365, Kluwer, Dordrecht, 1992, 185–194 | MR | Zbl

[3] L. Zalcman, Supplementary bibliography to “A bibliographic survey of the Pompeiu problem”, Radon transform and tomography (South Hadley, MA, USA, 2000), Contemp. Math., 278, Amer. Math. Soc., Providence, RI, 2001, 69–74 | MR | Zbl

[4] I. Netuka, J. Veselý, “Mean value property and harmonic functions”, Classical and modern potential theory and applications (Chateau de Bonas, France, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 430, Kluwer, Dordrecht, 1994, 359–398 | MR | Zbl

[5] V. V. Volchkov, Vit. V. Volchkov, “Ekstremalnye zadachi integralnoi geometrii”, Matematika segodnya, 1, no. 12, Nauchno-metodicheskii tsentr vysshego obrazovaniya, Kiev, 2001, 51–79 | MR | Zbl

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

[7] Vit. V. Volchkov, “O funktsiyakh s nulevymi sharovymi srednimi na kompleksnykh giperbolicheskikh prostranstvakh”, Matem. zametki, 68:4 (2000), 504–512 | MR | Zbl

[8] Vit. V. Volchkov, “Uniqueness theorems for mean periodic functions on complex hyperbolic spaces”, Anal. Math., 28:1 (2002), 61–76 | DOI | MR | Zbl

[9] Vit. V. Volchkov, “O funktsiyakh s nulevymi sharovymi srednimi na kvaternionnom giperbolicheskom prostranstve”, Izv. RAN. Ser. matem., 66:5 (2002), 3–32 | MR | Zbl

[10] Vit. V. Volchkov, “Teoremy edinstvennosti dlya periodicheskikh v srednem funktsii na kvaternionnom giperbolicheskom prostranstve”, Matem. zametki, 74:1 (2003), 32–40 | MR | Zbl

[11] V. V. Volchkov, “Teoremy o sharovykh srednikh na simmetricheskikh prostranstvakh”, Matem. sb., 192:9 (2001), 17–38 | MR | Zbl

[12] V. V. Volchkov, “Lokalnaya teorema o dvukh radiusakh na simmetricheskikh prostranstvakh”, Dokl. RAN, 381:6 (2001), 727–731 | MR | Zbl

[13] Vit. V. Volchkov, “Sfericheskie srednie na gruppe Geizenberga i simmetricheskikh prostranstvakh”, Dopovidi NAN Ukraïni. Matem. Prirodozn. Tekhn. Nauki, 3 (2005), 7–11 | MR | Zbl

[14] S. Khelgason, Gruppy i geometricheskii analiz, Integralnaya geometriya, invariantnye differentsialnye operatory i sfericheskie funktsii, Mir, M., 1987 ; S. Helgason, Groups and geometric analysis, Integral geometry, invariant differential operators, and spherical functions, Pure Appl. Math., 113, Academic Press, Orlando, FL, 1984 | MR | MR | Zbl

[15] T. Koornwinder, “A new proof of a Paley–Wiener type theorem for the Jacobi transform”, Ark. Mat., 13:1 (1975), 145–159 | DOI | MR | Zbl

[16] S. Khelgason, Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964 ; S. Helgason, Differential geometry and symmetric spaces, Pure Appl. Math., 12, Academic Press, New York–London, 1962 | Zbl | MR | Zbl

[17] A. L. Besse, Mnogoobraziya s zamknutymi geodezicheskimi, Mir, M., 1981 ; A. L. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb., 93, Springer-Verlag, Berlin–Heidelberg–New York, 1978 | MR | MR | Zbl

[18] Sh. Kobayasi, K. Nomidzu, Osnovy differentsialnoi geometrii, t. II, Nauka, M., 1981 ; Sh. Kobayashi, K. Nomizu, Foundations of differential geometry.Vol. II, Wiley-Intersci., New York–London–Sydney, 1969 | Zbl | MR | Zbl

[19] G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Stud., 78, Princeton Univ. Press, Princeton, NJ; Univ. Tokyo Press, Tokyo, 1973 | MR | Zbl

[20] Vit. V. Volchkov, Novye rezultaty v geometricheskom analize, Izd-vo Donetskogo nats. un-ta, Donetsk, 2004

[21] R. T. Smith, “The spherical representations of groups transitive on $\mathbb S^n$”, Indiana Univ. Math. J., 24:4 (1974), 307–325 | DOI | MR | Zbl

[22] N. Ya. Vilenkin, Spetsialnye funktsii i teoriya predstavlenii grupp, 2-e izd., Nauka, M., 1991 ; N. Ya. Vilenkin, Special functions and the theory of group representations, Transl. Math. Monogr., 22, Amer. Math. Soc., Providence, RI, 1968 | MR | Zbl | MR | Zbl

[23] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. I: Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1973 ; A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, vol. I, McGraw-Hill Book Co., New York–Toronto–London, 1953 | MR | MR | Zbl

[24] M. M. Postnikov, Lektsii po geometrii. Semestr V. Gruppy i algebry Li, Nauka, M., 1982 ; M. M. Postnikov, Lectures in geometry. SemesterV. Lie groups and Lie algebras, Mir, M., 1986 | MR | Zbl | MR

[25] L. V. Alfors, Preobrazovanie Mebiusa v mnogomernom prostranstve, Sovremennaya matematika: vvodnye kursy, Mir, M., 1986 ; L. V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures Math., Minneapolis, MN, Univ. of Minnesota, School of Math., 1981 | MR | Zbl | MR | Zbl

[26] U. Rudin, Teoriya funktsii v edinichnom share iz $\mathbb C^n$, Mir, M., 1984 ; W. Rudin, Function theory in the unit ball of $\mathbb C^n$, Grundlehren Math. Wiss., 241, Springer-Verlag, New York–Berlin, 1980 | MR | Zbl | MR | Zbl

[27] R. Askey, J. Fitch, “Integral representations for Jacobi polynomials and some applications”, J. Math. Anal. Appl, 26:2 (1969), 411–437 | DOI | MR | Zbl

[28] A. O. Gelfond, Ischislenie konechnykh raznostei, Nauka, M., 1967 ; A. O. Gel'fond, Calculus of finite differences, Hindustan Publ. Corp., Delhi, 1971 | MR | Zbl | MR | Zbl

[29] E. Ya. Riekstynsh, Asimptoticheskie razlozheniya integralov, t. 1, Zinatne, Riga, 1974 | MR | Zbl

[30] M. El Harchaoui, “Inversion de la transformation de Pompeiu locale dans les espaces hyperboliques réel et complexe: Cas des deux boules”, J. Anal. Math., 67 (1995), 1–37 | DOI | MR | Zbl

[31] M. Berkani, M. El Harchaoui, R. Gay, “Inversion de la transformation de Pompéiu locale dans l'espace hyperbolique quaternionique – cas des deux boules”, Complex Variables Theory Appl., 43:1 (2000), 29–57 | MR | Zbl

[32] F. Ion, Ploskie volny i sfericheskie srednie v primenenii k differentsialnym uravneniyam s chastnymi proizvodnymi, IL, M., 1958; F. John, Plane waves and spherical means applied to partial differential equations, Dover, Mineola, NY, 2004 | MR | Zbl

[33] V. V. Volchkov, Vit. V. Volchkov, Analogues of the local two-radii theorem on noncompact Riemannian symmetric spaces of arbitrary rank, Izd-vo Donetskogo nats. un-ta, Donetsk, 2004

[34] C. A. Berenstein, L. Zalcman, “Pompeiu's problem on symmetric spaces”, Comment. Math. Helv., 55:1 (1980), 593–621 | DOI | MR | Zbl

[35] P. K. Suetin, Klassicheskie ortogonalnye mnogochleny, Nauka, M., 1979 | MR | Zbl