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

For an integer $r\ge0$, we prove the $r$th order Reshetnyak formula for the ray transform of rank $m$ symmetric tensor fields on $\mathbb{R}^n$. Certain differential operators $A^{(m,r,l)}\ (0\le l\le r)$ on the sphere $\mathbb{S}^{n-1}$ are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any $r$ although the volume of calculations grows fast with $r$. The algorithm is realized for small values of $r$ and Reshetnyak formulas of orders $0,1,2$ are presented in an explicit form.