Local two-radii theorem in symmetric spaces

V. V. Volchkov

Sbornik. Mathematics, Tome 198 (2007) no. 11, pp. 1553-1577

Résumé

Various classes of functions on a non-compact rank-one Riemannian symmetric space $X$ with vanishing integrals over all balls of fixed radius are studied. A description in the form of a series in hypergeometric functions is obtained for such classes and a uniqueness theorem is proved. This makes it possible to establish the local two-radii theorem in $X$ in a definitive form. Bibliography: 45 titles.

Bibliographie

[1] L. Zalcman, “Analyticity and the Pompeiu problem”, Arch. Rational Mech. Anal., 47:3 (1972), 237–254 | DOI | 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] 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 | MR | Zbl

[4] L. Zalcman, “Mean values and differential equations”, Israel. J. Math., 14 (1973), 339–352 | DOI | MR | Zbl

[5] C. A. Berenstein, L. Zalcman, “Pompeiu's problem on spaces of constant curvature”, J. Analyse Math., 30 (1976), 113–130 | DOI | MR | Zbl

[6] L. Zalcman, “Offbeat integral geometry”, Amer. Math. Monthly, 87:3 (1980), 161–175 | DOI | MR | Zbl

[7] M. L. Agranovskiĭ, “The Fourier transform on $SL_2(R)$ and theorems of Morera type”, Soviet Math. Dokl., 19:6 (1978), 1522–1526 | MR | Zbl

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

[9] S. Thangavelu, “Spherical means and CR functions on the Heisenberg group”, J. Anal. Math., 63 (1994), 255–286 | DOI | MR | Zbl

[10] M. Agranovsky, C. Berenstein, D.-Ch. Chang, D. Pascuas, “Théorèmes de Morera et Pompeiu pour le groupe de Heisenberg”, C. R. Acad. Sci. Paris Sér. I Math., 315:6 (1992), 655–658 | MR | Zbl

[11] M. Agranovsky, C. Berenstein, D.-Ch. Chang, “Morera theorem for holomorphic $H^p$ spaces in the Heisenberg group”, J. Reine Angew. Math., 443 (1993), 49–89 | MR | Zbl

[12] J. M. Cohen, M. A. Picardello, “The 2-circle and 2-disk problems on trees”, Israel J. Math., 64:1 (1988), 73–86 | DOI | MR | Zbl

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

[14] C. A. Berenstein, R. Gay, “A local version of the two-circles theorem”, Israel J. Math., 55:3 (1986), 267–288 | DOI | MR | Zbl

[15] C. A. Berenstein, R. Gay, “Le problème de Pompeiu local”, J. Analyse Math., 52 (1989), 133–166 | DOI | MR | Zbl

[16] 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

[17] V. V. Volchkov, “A definitive version of the local two-radii theorem”, Sb. Math., 186:6 (1995), 783–802 | DOI | MR | Zbl

[18] V. V. Volchkov, “A definitive version of the local two-radii theorem on hyperbolic spaces”, Izv. Math., 65:2 (2001), 207–229 | DOI | MR | Zbl

[19] V. V. Volchkov, “On a problem of Zalcman and its generalizations”, Math. Notes, 53:2 (1993), 134–138 | DOI | MR | Zbl

[20] V. V. Volchkov, “Solution of the support problem for several function classes”, Sb. Math., 188:9 (1997), 1279–1294 | DOI | MR | Zbl

[21] V. V. Volchkov, “Injectivity sets for the Radon transform over a sphere”, Izv. Math., 63:3 (1999), 481–493 | DOI | MR

[22] V. V. Volchkov, “Injectivity sets of the Pompeiu transform”, Sb. Math., 190:11 (1999), 1607–1622 | DOI | MR | Zbl

[23] V. V. Volchkov, “Extremal problems on Pompeiu sets”, Sb. Math., 189:7 (1998), 955–976 | DOI | MR | Zbl

[24] V. V. Volchkov, “Extremal problems on Pompeiu sets. II”, Sb. Math., 191:5 (2000), 619–632 | DOI | MR | Zbl

[25] V. V. Volchkov, “New theorems on the mean for solutions of the Helmholtz equation”, Russian Acad. Sci. Sb. Math., 79:2 (1994), 281–286 | DOI | MR | Zbl

[26] V. V. Volchkov, “Extremal cases of the Pompeiu problem”, Math. Notes, 59:5 (1996), 482–489 | DOI | MR | Zbl

[27] V. V. Volchkov, “On polyhedra with the local Pompeiu property”, Russian Acad. Sci. Dokl. Math., 62:1 (2000), 69–71 | MR | Zbl

[28] V. V. Volchkov, “On functions with zero integrals over ellipsoids”, Russian Acad. Sci. Dokl. Math., 63:1 (2001), 25–27 | MR | MR | Zbl

[29] V. V. Volchkov, “Uniqueness theorems for multiple lacunary trigonometric series”, Math. Notes, 51:6 (1992), 550–552 | DOI | MR | Zbl

[30] V. V. Volchkov, “Uniqueness theorems for multiple lacunary trigonometric series”, Ukrainian Math. J., 50:11 (1998), 1686–1691 | DOI | MR | Zbl

[31] V. V. Volchkov, “Approximation of functions on bounded domains in $\mathbb R^n$ by linear combinations of shifts”, Russian Acad. Sci. Dokl. Math., 49:1 (1994), 160–162 | MR | Zbl

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

[33] 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

[34] S. Helgason, Differential geometry and symmetric spaces, Pure Appl. Math., 12, Academic Press, New York–London, 1962 | MR | Zbl | Zbl

[35] S. Helgason, Differential geometry, Lie group and symmetric spaces, Pure Appl. Math., 80, Academic Press, New York–London, 1978 | MR | Zbl

[36] S. Helgason, Geometric analysis on symmetric spaces, Math. Surveys Monogr., 39, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl

[37] Vit. V. Volchkov, “Teoremy o sharovykh srednikh na kompleksnykh giperbolicheskikh prostranstvakh”, Dopovidi NAN Ukraїni, 4 (2000), 7–10 | MR | Zbl

[38] Vit. V. Volchkov, “An analog of the Poincaré model for a quaternion hyperbolic space”, Ukrainian Math. J., 53:10 (2001), 1618–1625 | DOI | MR | Zbl

[39] N. Ya. Vilenkin, Special functions and the theory of group representations, Transl. Math. Monogr., 22, Amer. Math. Soc., Providence, RI, 1968 | MR | MR | Zbl | Zbl

[40] M. Flensted-Jensen, “Paley–Wiener type theorems for a differential operator connected with symmetric spaces”, Ark. Mat., 10:1 (1972), 143–162 | DOI | MR | Zbl

[41] 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

[42] 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

[43] B. Ya. Levin, Distribution of zeros of entire functions, Amer. Math. Soc., Providence, RI, 1964 | MR | MR | Zbl | Zbl

[44] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations, Clarendon Press, Oxford, 1946 | MR | MR | Zbl | Zbl

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