A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity

References

[1] A. Baernstein, II and M. Loss, Some conjectures about L^p norms of k-plane transforms. Rend. Sem. Mat. Fis. Milano67 (1997), 9–26.10.1007/BF02930488Suche in Google Scholar

[2] J. Bennett, N. Bez, T.C. Flock, S. Gutiérrez, and M. Iliopoulou, A sharp k-plane Strichartz inequality for the Schrödinger equation. Trans. Amer. Math. Soc. 370 (2018), 5617—5633.10.1090/tran/7309Suche in Google Scholar

[3] W. Blaschke, Integralgeometrie 2: Zu Ergebnissen von M.W. Crofton. Bull. Math. Soc. Roum. Sci. 37 (1935), 3–11.Suche in Google Scholar

[4] M. Christ, Estimates for the k-plane transform. Indiana Univ. Math. J. 33 (1984), 891—910.10.1512/iumj.1984.33.33048Suche in Google Scholar

[5] S. Dann, G. Paouris and P. Pivovarov, Bounding marginal densities via affine isoperimetry. Proc. Lond. Math. Soc. 113, No 3 (2016), 140–162.10.1112/plms/pdw026Suche in Google Scholar

[6] A. Drouot, Sharp constant for a k-plane transform inequality. Anal. PDE7 (2014), 1237—1252.10.2140/apde.2014.7.1237Suche in Google Scholar

[7] S.W. Drury, Generalizations of Riesz potentials and L^p estimates for certain k-plane transforms. Illinois J. Math. 28 (1984), 495–512.Suche in Google Scholar

[8] J. Faraut, and G. Travaglini, Bessel functions associated with representations of formally real Jordan algebras. J. of Funct. Analysis71 (1987), 123–141.10.1016/0022-1236(87)90019-XSuche in Google Scholar

[9] T.C. Flock, Uniqueness of extremizers for an endpoint inequality of the k-plane transform. J. Geom. Anal. 26 (2016), 570—602.10.1007/s12220-015-9563-0Suche in Google Scholar

[10] P.J. Forrester, Matrix polar decomposition and deneralisations of the Blaschke-Petkantschin formula in integral geometry. ArXiv Preprint, arXiv:1701.04505, 2017.Suche in Google Scholar

[11] F.R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea Publ. Company, New York, 1959.Suche in Google Scholar

[12] R.J. Gardner, The dual Brunn-Minkowski theory for bounded Borel sets: dual affine quermassintegrals and inequalities. Adv. Math. 216 (2007), 358–386.10.1016/j.aim.2007.05.018Suche in Google Scholar

[13] S.S. Gelbart, Fourier Analysis on Matrix Space. Memoirs of the Amer. Math. Soc., 108, AMS, Providence, RI, 1971.10.1090/memo/0108Suche in Google Scholar

[14] C. Herz, Bessel functions of matrix argument. Ann. of Math. 61 (1955), 474–523.10.2307/1969810Suche in Google Scholar

[15] E.B.V. Jensen, Local Stereology. Advanced Ser. on Statistical Science & Applied Probability, 5, World Scientific Publishing Co., River Edge, NJ, 1998.10.1142/2926Suche in Google Scholar

[16] S.P. Khekalo, Riesz potentials in the space of rectangular matrices and iso-Huygens deformations of the Cayley-Laplace operator. Doklady Mathematics63, No 1 (2001), 35–37.Suche in Google Scholar

[17] R.E. Miles, Isotropic random simplices. Advances in Appl. Probability3 (1971), 353–382.10.2307/1426176Suche in Google Scholar

[18] E. Milman, Generalized intersection bodies. J. Funct. Anal. 240 (2006), 530—567.10.1016/j.jfa.2006.04.004Suche in Google Scholar

[19] S. Moghadasi, Polar decomposition of the k-fold product of Lebesgue measure on ℝⁿ. Bull. Aust. Math. Soc. 85 (2012), 315–324.10.1017/S0004972711003273Suche in Google Scholar

[20] R.J. Muirhead, Aspects of Multivariate Statistical Theory. John Wiley & Sons, New York, 1982.10.1002/9780470316559Suche in Google Scholar

[21] B. Petkantschin, Integralgeometrie 6. Zusammenhänge zwischen den Dichten der linearen Unterräume im n-dimensionalen Raum. Abh. Math. Semin. Univ. Hambg. 11, Issue 1 (1935), 249–310.10.1007/BF02940729Suche in Google Scholar

[22] M. Raïs, Distributions homogènes sur des espaces de matrices. Bull. Soc. Math. France, Mem. 30 (1972), 3–109.10.24033/msmf.57Suche in Google Scholar

[23] B. Rubin, Radon transforms on affine Grassmannians. Trans. Amer. Math. Soc. 356 (2004), 5045–5070.10.1090/S0002-9947-04-03508-1Suche in Google Scholar

[24] B. Rubin, Riesz potentials and integral geometry in the space of rectangular matrices. Advances in Math. 205 (2006), 549–598.10.1016/j.aim.2005.08.001Suche in Google Scholar

[25] L. Santalo, Integral Geometry and Geometric Probability, Cambridge University Press, Cambridge, 2004.10.1017/CBO9780511617331Suche in Google Scholar

[26] R. Schneider and W. Weil, Integralgeometrie (In German). Teubner Skripten zur Mathematischen Stochastik, B. G. Teubner Stuttgart, 1992.10.1007/978-3-322-84824-6Suche in Google Scholar

[27] R. Schneider and W. Weil, Stochastic and Integral Geometry. Springer, Berlin-Heidelberg, 2008.10.1007/978-3-540-78859-1Suche in Google Scholar

[28] E.M. Stein, Analysis in matrix spaces and some new representations of SL(N,C). Ann. of Math. 86 (1967), 461–490.10.2307/1970611Suche in Google Scholar

[29] G. Zhang, Radon transform on real, complex, and quaternionic Grassmannians. Duke Math. J. 138 (2007), 137–160.10.1215/S0012-7094-07-13814-6Suche in Google Scholar