Dual Radon transforms on affine Grassmann manifolds

Authors:
Fulton B. Gonzalez and Tomoyuki Kakehi

Journal:
Trans. Amer. Math. Soc. **356** (2004), 4161-4180

MSC (2000):
Primary 44A12; Secondary 43A85

DOI:
https://doi.org/10.1090/S0002-9947-04-03471-3

Published electronically:
April 16, 2004

MathSciNet review:
2058842

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Fix , and let and denote the affine Grassmann manifolds of - and -planes in . We investigate the Radon transform associated with the inclusion incidence relation. For the generic case and , we will show that the range of this transform is given by smooth functions on annihilated by a system of Pfaffian type differential operators. We also study aspects of the exceptional case .

**[G1]**F. Gonzalez,*Radon transforms on Grassmann manifolds*, J. Funct. Anal.**71**(1987), 339-362. MR**89a:53081****[G2]**F. Gonzalez,*Bi-invariant differential operators on the Euclidean motion group and applications to generalized Radon transforms*, Ark. Mat.**26**(1988), 191-204. MR**92c:58144****[G3]**F. Gonzalez,*Invariant differential operators and the range of the Radon -plane transform*, Math. Ann.**287**(1990), 627-635. MR**92a:58141****[G4]**F. Gonzalez,*On the range of the Radon transform on Grassmann manifolds*, preprint.**[GH]**F. Gonzalez and S. Helgason,*Invariant differential operators on Grassmann manifolds*, Adv. in Math.**60**(1986), 81-91. MR**87j:22015****[GK]**F. Gonzalez and T. Kakehi,*Pfaffian Systems and Radon Transforms on Affine Grassmann Manifolds*, Math. Ann.**326**(2003), 237-273.**[Gr]**E. Grinberg,*Radon transforms on higher rank Grassmannians*, J. Diff. Geom.**24**(1986), 53-68. MR**87m:22023****[GrRu]**E. Grinberg and B. Rubin,*Radon inversion on Grassmannians via Garding-Gindikin fractional integrals*, to appear in Annals of Mathematics.**[GR]**I. S. Gradshteyn and I. M. Ryzhik,*Table of Integrals, Series, and Products*, Academic Press, 1980.**[H1]**S. Helgason,*The Radon transform on Euclidean spaces, two-point homogeneous spaces, and Grassmann manifolds*, Acta Math.**113**(1965), 153-180. MR**30:2530****[H2]**S. Helgason,*Groups and Geometric Analysis*, Academic Press, Orlando, 1984. MR**86c:22017****[H3]**S. Helgason,*Geometric Analysis on Symmetric Spaces*, AMS, Mathematical Surveys and Monographs,**39**Providence, 1994. MR**96h:43009****[H4]**S. Helgason,*The Radon Transform*, Second edition, Progress in Mathematics,**5**, Birkhäuser, Boston, 1999. MR**2000m:44003****[J]**F. John,*The ultrahyperbolic differential equation with independent variables*, Duke Math. J.**4**(1938), 300-322.**[K]**T. Kakehi,*Integral geometry on Grassmann manifolds and calculus of invariant differential operators*, J. Funct. Anal.**168**(1999), 1-45. MR**2000k:53069****[KN]**S. Kobayashi and K. Nomizu,*Foundations of differential geometry I, II*, Wiley, New York, 1963 and 1969. MR**38:6501****[P]**E.E. Petrov,*The Radon transform in spaces of matrices and in Grassmann manifolds*, Dokl. Akad. Nauk SSSR,**177**(1967), 1504-1507. MR**36:7095****[Ri]**F. Richter,*On the -dimensional Radon transform of rapidly decreasing functions*, Lecture Notes in Math. No. 1209, Berlin, New York, 1986. MR**88a:53071****[Ru1]**B. Rubin,*Inversion and characterization of the hemispherical transform*, J. D'Analyse Math.**77**(1999), 105-128. MR**2001m:44004****[Ru2]**B. Rubin,*Radon transforms on affine Grassmannians*, The Hebrew University of Jerusalem, (2003) preprint.**[So]**D. Solmon,*Asymptotic formulas for the dual Radon transform*, Math. Zeitschr.**195**(1987), 321-343. MR**88i:44006****[St]**R. Strichartz,*Harmonic analysis on Grassmann bundles*, Trans. Amer. Math. Soc.**296**(1986), 387-409. MR**88b:43006**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
44A12,
43A85

Retrieve articles in all journals with MSC (2000): 44A12, 43A85

Additional Information

**Fulton B. Gonzalez**

Affiliation:
Department of Mathematics, Tufts University, Medford, Massachusetts 02155-7049

Email:
fulton.gonzalez@tufts.edu

**Tomoyuki Kakehi**

Affiliation:
Institute of Mathematics, University of Tsukuba, Ibaraki, Japan 305-8571

Email:
kakehi@math.tsukuba.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-04-03471-3

Keywords:
Radon transform,
Grassmannian,
Pfaffian systems

Received by editor(s):
November 26, 2002

Received by editor(s) in revised form:
May 1, 2003, and July 17, 2003

Published electronically:
April 16, 2004

Article copyright:
© Copyright 2004
American Mathematical Society