Helgason-Marchaud inversion formulas for Radon transforms

Abstract:

Let $X$ be either the hyperbolic space $\mathbb {H} ^{n}$ or the unit sphere $S^{n}$, and let $\Xi$ be the set of all $k$-dimensional totally geodesic submanifolds of $X, 1 \le k \le n-1$. For $x \in X$ and $\xi \in \Xi$, the totally geodesic Radon transform $f(x) \to \hat f(\xi )$ is studied. By averaging $\hat f(\xi )$ over all $\xi$ at a distance $\theta$ from $x$, and applying Riemann-Liouville fractional differentiation in $\theta$, S. Helgason has recovered $f(x)$. We show that in the hyperbolic case this method blows up if $f$ does not decrease sufficiently fast. The situation can be saved if one employs Marchaud’s fractional derivatives instead of the Riemann-Liouville ones. New inversion formulas for $\hat f(\xi ), f \in L^{p}(X)$, are obtained.