The Radon transform \documentclass[12pt]{minimal}
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\begin{document}$ R $\end{document} maps a function \documentclass[12pt]{minimal}
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\begin{document}$ f $\end{document} on \documentclass[12pt]{minimal}
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\begin{document}$ {}^{n} $\end{document}
to the family of the integrals of \documentclass[12pt]{minimal}
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\begin{document}$ f $\end{document} over all hyperplanes. The classical
Reshetnyak formula (also called the Plancherel formula for the Radon transform) states that
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\begin{document}$ \|f\|_{L^{2}({}^{n})}=\|Rf\|_{H^{(n-1)/2}_{(n-1)/2}({𝕊}^{n-1}\times{})} $\end{document},
where
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\begin{document}$ \|\cdot\|_{H^{(n-1)/2}_{(n-1)/2}({𝕊}^{n-1}\times{})} $\end{document}
is some special norm. The formula extends the Radon transform to the bijective Hilbert space isometry
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\begin{document}$ R:L^{2}({}^{n})\rightarrow H^{(n-1)/2}_{(n-1)/2,e}({𝕊}^{n-1}\times{}) $\end{document}.
Given reals \documentclass[12pt]{minimal}
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\begin{document}$ r $\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$ s $\end{document}, and \documentclass[12pt]{minimal}
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\begin{document}$ t>-n/2 $\end{document}, we introduce the Sobolev type spaces
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\begin{document}$ H^{(r,s)}_{t}({}^{n}) $\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$ H^{(r,s)}_{t,e}({𝕊}^{n-1}\times{}) $\end{document}
and prove the version of the Reshetnyak formula:
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\begin{document}$ \|f\|_{H^{(r,s)}_{t}({}^{n})}=\|Rf\|_{H^{(r,(s+n-1)/2)}_{t+(n-1)/2}({𝕊}^{n-1}\times{})} $\end{document}.
The formula extends the Radon transform to the bijective Hilbert space isometry
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\begin{document}$ R:H^{(r,s)}_{t}({}^{n})\rightarrow H^{(r,s+(n-1)/2)}_{t+(n-1)/2,e}({𝕊}^{n-1}\times{}) $\end{document}.
If \documentclass[12pt]{minimal}
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\begin{document}$ r\geq 0 $\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$ s\geq 0 $\end{document} are integers then \documentclass[12pt]{minimal}
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\begin{document}$ H^{(r,s)}_{0,e}({𝕊}^{n-1}\times{}) $\end{document}
consists of the even functions \documentclass[12pt]{minimal}
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\begin{document}$ \varphi(\xi,p) $\end{document}
with square integrable derivatives of order \documentclass[12pt]{minimal}
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\begin{document}$ \leq r $\end{document} with respect to \documentclass[12pt]{minimal}
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\begin{document}$ \xi $\end{document} and order \documentclass[12pt]{minimal}
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\begin{document}$ \leq s $\end{document} with respect to \documentclass[12pt]{minimal}
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\begin{document}$ p $\end{document}.
