Radon transform on Sobolev spaces

The Radon transform $R$ maps a function $f$ on ${\Bbb R}^n$ to the family of the integrals of $f$ over all hyperplanes. The classical Reshetnyak formula (also called the Plancherel formula for the Radon transform) states that $\|f\|_{L^2({\Bbb R}^n)}=\|Rf\|_{H^{(n-1)/2}_{(n-1)/2}({\Bbb S}^{n-1}\times{\Bbb R})}$, where $\|\cdot\|_{H^{(n-1)/2}_{(n-1)/2}({\Bbb S}^{n-1}\times{\Bbb R})}$ is some special norm. The formula extends the Radon transform to the bijective Hilbert space isometry $R:L^2({\Bbb R}^n)\rightarrow H^{(n-1)/2}_{(n-1)/2,e}({\Bbb S}^{n-1}\times{\Bbb R})$. Given reals $r$, $s$, and $t>-n/2$, we introduce the Sobolev type spaces $H^{(r,s)}_t({\Bbb R}^n)$ and $H^{(r,s)}_{t,e}({\Bbb S}^{n-1}\times{\Bbb{R}})$ and prove the version of the Reshetnyak formula: $\|f\|_{H^{(r,s)}_t({\Bbb R}^n)}=\|Rf\|_{H^{(r,(s+n-1)/2)}_{t+(n-1)/2}({\Bbb S}^{n-1}\times{\Bbb R})}$. The formula extends the Radon transform to the bijective Hilbert space isometry $R:H^{(r,s)}_t({\Bbb R}^n)\rightarrow H^{(r,s+(n-1)/2)}_{t+(n-1)/2,e}({\Bbb S}^{n-1}\times{\Bbb R})$. If $r\ge0$ and $s\ge0$ are integers then $H^{(r,s)}_{0,e}({\Bbb S}^{n-1}\times {\Bbb{R}})$ consists of the even functions $\varphi(\xi,p)$ with square integrable derivatives of order $\le r$ with respect to $\xi$ and order $\le s$ with respect to $p$.