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

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 $\left \| {R_{1,n}}f\right \|_{{L^q}({M_\phi }({{\mathbf {R}}^n}),d\mu )} \leq C\left \| f\right \| _{{L^p}({{\mathbf {R}}^n})}$ 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.