  We consider different norms for the Radon transform $Rf$ of a function $f$
and investigate under which conditions they can be estimated from above or
below by some standard norms for $f$. We define Fourier-based norms for $Rf$
that can be related to Bessel-potential space norms for $f$. Furthermore, we
define a variant of a total-variation norm for $Rf$ and provide conditions
under which it is equivalent to the total-variation norm of $f$. As an
illustration of potential applications of these results, we propose a novel
nonlinear backprojection method for inverting the Radon transform.
