Radon Transforms over Horospheres in Real Hyperbolic Space
The horospherical Radon transform integrates functions on the n-dimensional real
hyperbolic space over d-dimensional horospheres, where d is a fixed integer, $1\le d\le n-1$.
Using the tools of real analysis, we obtain sharp existence conditions and explicit inversion
formulas for these transforms acting on smooth functions and functions belonging to $L^p$. The
case d = n-1 agrees with the classical Gelfand-Graev transform which was studied before in
terms of the distribution theory on rapidly decreasing smooth functions. The results for
$L^p$-functions and the case d < n-1 are new. This is a joint work with William O. Bray.