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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

$ L\sp p$ estimates for the X-ray transform restricted to line complexes of Kirillov type


Author: Hann Tzong Wang
Journal: Trans. Amer. Math. Soc. 332 (1992), 793-821
MSC: Primary 53C65; Secondary 44A12, 92C55
MathSciNet review: 1052912
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Abstract: Let there be given a piecewise continuous rectifiable curve $ \phi :{\mathbf{R}} \to {{\mathbf{R}}^n}$. Let $ {G_{1,n}}({M_{1,n}})$ be the usual Grassmannian (bundle) in $ {{\mathbf{R}}^n}$. Define an $ n$-dimensional submanifold $ {M_\phi }({{\mathbf{R}}^n})$ of $ {M_{1,n}}$ as the set of all copies of $ {G_{1,n}}$ along the curve $ \phi $. Following Kirillov, we know that a nice function $ f(x)$ can be recovered from its X-ray transform $ {R_{1,n}}f$ on $ {M_\phi }({{\mathbf{R}}^n})$ if and only if the curve $ \phi $ intersects almost every affine hyperplane. Define a measure on $ {M_\phi }({{\mathbf{R}}^n})$ by $ d\mu = d{\mu _x}(\pi )d\lambda (x)$, where $ d{\mu _x}$ is the probability measure on $ {M_{1,n}}$ carried by the set of lines passing through the point $ x$ and invariant under the stabilizer of $ x$ in $ O(n)$ and $ d\lambda $ is the usual measure on $ \phi $. We show that, if $ n > 2$ and $ \phi $ is unbounded, then $ \left\Vert {R_{1,n}}f\right\Vert _{{L^q}({M_\phi }({{\mathbf{R}}^n}),d\mu )} \leq C\left\Vert f\right\Vert _{{L^p}({{\mathbf{R}}^n})}$ if and only if $ p = q = n - 1$ and $ \phi $ is line-like, that is, $ \lambda (\phi \cap B(0;R)) = O(R)$. This result gives a classification of Kirillov line complexes in terms of $ {L^p}$ estimates.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1052912-6
Article copyright: © Copyright 1992 American Mathematical Society