Plancherel's formula for the Radon transform in the flexible scale of Sobolev Hilbert spaces

Abstract

The mathematical theory of the Radon transform with technical implementation in tomographs has an unlimited range of applications, among which the widest are medical ones.

In the aspect of studies of the Radon transformation, the key role is played by the ratio:

For any  $\beta >0$ \textit{there are exist positive constants $c\left(\beta ,\, s\right)$ и $C\left(\beta ,\, s\right)$, such that $f\in C_{0}^{\infty } \left(\Omega _{s} \right)$

\[c\left(\beta, s\right){\left\|f\right\|}_{W^{\beta }_2\left(\Omega_s\right)}\le {\left\|Rf\right\|}_{W^{\beta +\frac{\left(s-1\right)}{2}}\left(Z=S^{s-1}\times R^1\right)}\le C\left(\beta, s\right){\left\|f\right\|}_{W^{\beta}_2\left(\Omega_s\right)}.\]

This article is devoted to extending this equivalence in the form of equality to the far-reaching case of Sobolev spaces.