Radon transforms over lower-dimensional horospheres in real hyperbolic space

Bray, William; Rubin, Boris

doi:10.1090/tran/7666

Radon transforms over lower-dimensional horospheres in real hyperbolic space
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by William O. Bray and Boris Rubin PDF

Trans. Amer. Math. Soc. 372 (2019), 1091-1112 Request permission

Abstract:

We study horospherical Radon transforms that integrate functions on the $n$-dimensional real hyperbolic space over horospheres of arbitrary fixed dimension $1\le d\le n-1$. Exact existence conditions and new explicit inversion formulas are obtained for these transforms acting on smooth functions and functions belonging to $L^p$. The case $d=n-1$ agrees with the well-known Gelfand-Graev transform.

References

Carlos A. Berenstein and Enrico Casadio Tarabusi, An inversion formula for the horocyclic Radon transform on the real hyperbolic space, Tomography, impedance imaging, and integral geometry (South Hadley, MA, 1993) Lectures in Appl. Math., vol. 30, Amer. Math. Soc., Providence, RI, 1994, pp. 1–6. MR 1297561
William O. Bray, Aspects of harmonic analysis on real hyperbolic space, Fourier analysis (Orono, ME, 1992) Lecture Notes in Pure and Appl. Math., vol. 157, Dekker, New York, 1994, pp. 77–102. MR 1277819
William O. Bray and Boris Rubin, Inversion of the horocycle transform on real hyperbolic spaces via a wavelet-like transform, Analysis of divergence (Orono, ME, 1997) Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1999, pp. 87–105. MR 1731261
Bent Fuglede, An integral formula, Math. Scand. 6 (1958), 207–212. MR 105724, DOI 10.7146/math.scand.a-10545
I. M. Gel’fand, Integral geometry and its relation to the theory of representations, Russian Math. Surveys 15 (1960), no. 2, 143–151.
I. M. Gel′fand and M. I. Graev, Geometry of homogeneous spaces, representations of groups in homogeneous spaces and related questions of integral geometry. I, Trudy Moskov. Mat. Obšč. 8 (1959), 321–390; addendum 9 (1959), 562 (Russian). MR 0126719
I. M. Gel′fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5: Integral geometry and representation theory, Academic Press, New York-London, 1966. Translated from the Russian by Eugene Saletan. MR 0207913
Simon Gindikin, Integral geometry on hyperbolic spaces, Harmonic analysis and integral geometry (Safi, 1998) Chapman & Hall/CRC Res. Notes Math., vol. 422, Chapman & Hall/CRC, Boca Raton, FL, 2001, pp. 41–46. MR 1789143
S. G. Gindikin, Horospherical transform on symmetric Riemannian manifolds of noncompact type, Funktsional. Anal. i Prilozhen. 42 (2008), no. 4, 50–59, 111–112 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 42 (2008), no. 4, 290–297. MR 2492426, DOI 10.1007/s10688-008-0042-2
Simon Gindikin, Local inversion formulas for horospherical transforms, Mosc. Math. J. 13 (2013), no. 2, 267–280, 363 (English, with English and Russian summaries). MR 3134907, DOI 10.17323/1609-4514-2013-13-2-267-280
Fulton B. Gonzalez, Conical distributions on the space of flat horocycles, J. Lie Theory 20 (2010), no. 3, 409–436. MR 2743098
Fulton Gonzalez and Eric Todd Quinto, Support theorems for Radon transforms on higher rank symmetric spaces, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1045–1052. MR 1205492, DOI 10.1090/S0002-9939-1994-1205492-3
Sigurdur Helgason, The surjectivity of invariant differential operators on symmetric spaces. I, Ann. of Math. (2) 98 (1973), 451–479. MR 367562, DOI 10.2307/1970914
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Sigurdur Helgason, Geometric analysis on symmetric spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 2008. MR 2463854, DOI 10.1090/surv/039
Sigurdur Helgason, Integral geometry and Radon transforms, Springer, New York, 2011. MR 2743116, DOI 10.1007/978-1-4419-6055-9
Sigurdur Helgason, Support theorems for horocycles on hyperbolic spaces, Pure Appl. Math. Q. 8 (2012), no. 4, 921–927. MR 2959914, DOI 10.4310/PAMQ.2012.v8.n4.a4
J. Hilgert, A. Pasquale, and E. B. Vinberg, The dual horospherical Radon transform for polynomials, Mosc. Math. J. 2 (2002), no. 1, 113–126, 199 (English, with English and Russian summaries). MR 1900587, DOI 10.17323/1609-4514-2002-2-1-113-126
J. Hilgert, A. Pasquale, and E. B. Vinberg, The dual horospherical Radon transform as a limit of spherical Radon transforms, Lie groups and symmetric spaces, Amer. Math. Soc. Transl. Ser. 2, vol. 210, Amer. Math. Soc., Providence, RI, 2003, pp. 135–143. MR 2018358, DOI 10.1090/trans2/210/10
E. Kelly, Lower-dimensional horocycles and the Radon transforms on symmetric spaces, Preprint, 1974.
P. I. Lizorkin, Direct and inverse theorems in approximation theory for functions on a Lobachevskiĭ space, Trudy Mat. Inst. Steklov. 194 (1992), no. Issled. po Teor. Differ. Funktsiĭ Mnogikh Peremen. i ee Prilozh. 14, 120–147 (Russian); English transl., Proc. Steklov Inst. Math. 4(194) (1993), 125–151. MR 1289652
Mitsuo Morimoto, Sur les transformations horosphériques généralisées dans les espaces homogènes, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 65–83 (1966) (French). MR 209407
Mitsuo Morimoto, On the Radon transform, Sūgaku 20 (1968), 1–7 (Japanese). MR 252570
Boris Rubin, Helgason-Marchaud inversion formulas for Radon transforms, Proc. Amer. Math. Soc. 130 (2002), no. 10, 3017–3023. MR 1908925, DOI 10.1090/S0002-9939-02-06554-1
Boris Rubin, Inversion formulas for the spherical Radon transform and the generalized cosine transform, Adv. in Appl. Math. 29 (2002), no. 3, 471–497. MR 1942635, DOI 10.1016/S0196-8858(02)00028-3
Boris Rubin, Radon, cosine and sine transforms on real hyperbolic space, Adv. Math. 170 (2002), no. 2, 206–223. MR 1932329, DOI 10.1006/aima.2002.2074
Boris Rubin, On the Funk-Radon-Helgason inversion method in integral geometry, Geometric analysis, mathematical relativity, and nonlinear partial differential equations, Contemp. Math., vol. 599, Amer. Math. Soc., Providence, RI, 2013, pp. 175–198. MR 3202479, DOI 10.1090/conm/599/11908
Boris Rubin, Introduction to Radon transforms, Encyclopedia of Mathematics and its Applications, vol. 160, Cambridge University Press, New York, 2015. With elements of fractional calculus and harmonic analysis. MR 3410931
B. Rubin, Overdetermined transforms in integral geometry, Complex analysis and dynamical systems VI. Part 1, Contemp. Math., vol. 653, Amer. Math. Soc., Providence, RI, 2015, pp. 291–313. MR 3453082, DOI 10.1090/conm/653/13200
Boris Rubin, New inversion formulas for the horospherical transform, J. Geom. Anal. 27 (2017), no. 1, 908–946. MR 3606575, DOI 10.1007/s12220-016-9704-0
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
N. Ja. Vilenkin, Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by V. N. Singh. MR 0229863
N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions. Vol. 2, Mathematics and its Applications (Soviet Series), vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1993. Class I representations, special functions, and integral transforms; Translated from the Russian by V. A. Groza and A. A. Groza. MR 1220225, DOI 10.1007/978-94-017-2883-6