## Radon transforms on affine Grassmannians

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- by Boris Rubin PDF
- Trans. Amer. Math. Soc.
**356**(2004), 5045-5070 Request permission

## Abstract:

We develop an analytic approach to the Radon transform $\hat f (\zeta )=\int _{\tau \subset \zeta } f (\tau )$, where $f(\tau )$ is a function on the affine Grassmann manifold $G(n,k)$ of $k$-dimensional planes in $\mathbb {R}^n$, and $\zeta$ is a $k’$-dimensional plane in the similar manifold $G(n,k’), \; k’>k$. For $f \in L^p (G(n,k))$, we prove that this transform is finite almost everywhere on $G(n,k’)$ if and only if $\; 1 \le p < (n-k)/(k’ -k)$, and obtain explicit inversion formulas. We establish correspondence between Radon transforms on affine Grassmann manifolds and similar transforms on standard Grassmann manifolds of linear subspaces of $\mathbb {R}^{n+1}$. It is proved that the dual Radon transform can be explicitly inverted for $k+k’ \ge n-1$, and interpreted as a direct, “quasi-orthogonal" Radon transform for another pair of affine Grassmannians. As a consequence we obtain that the Radon transform and the dual Radon transform are injective simultaneously if and only if $k+k’ = n-1$. The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.## References

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## Additional Information

**Boris Rubin**- Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
- MR Author ID: 209987
- Email: boris@math.huji.ac.il
- Received by editor(s): May 13, 2003
- Received by editor(s) in revised form: September 11, 2003
- Published electronically: June 29, 2004
- Additional Notes: This work was supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**356**(2004), 5045-5070 - MSC (2000): Primary 44A12; Secondary 47G10
- DOI: https://doi.org/10.1090/S0002-9947-04-03508-1
- MathSciNet review: 2084410

Dedicated: Dedicated to Professor Lawrence Zalcman on the occasion of his 60th birthday