Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

Abstract

For an integer \begin{document}$ r\ge0 $\end{document}, we prove the \begin{document}$ r^{\mathrm{th}} $\end{document} order Reshetnyak formula for the ray transform of rank \begin{document}$ m $\end{document} symmetric tensor fields on \begin{document}$ {{\mathbb R}}^n $\end{document}. Roughly speaking, for a tensor field \begin{document}$ f $\end{document}, the order \begin{document}$ r $\end{document} refers to \begin{document}$ L^2 $\end{document}-integrability of higher order derivatives of the Fourier transform \begin{document}$ \widehat f $\end{document} over spheres centered at the origin. Certain differential operators \begin{document}$ A^{(m,r,l)}\ (0\le l\le r) $\end{document} on the sphere \begin{document}$ {{\mathbb S}}^{n-1} $\end{document} are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any \begin{document}$ r $\end{document} although the volume of calculations grows fast with \begin{document}$ r $\end{document}. The algorithm is realized for small values of \begin{document}$ r $\end{document} and Reshetnyak formulas of orders \begin{document}$ 0,1,2 $\end{document} are presented in an explicit form.