Normal operators for momentum ray transforms, II: Saint Venant operator

The momentum ray transform $I_m^k$ integrates a rank $m$ symmetric tensor field $f$ on ${\mathbb{R}}^n$ over lines with the weight $t^k$, $I_m^{k}f(x,\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{t}^k⟨f(x+t\xi ),{\xi }^m⟩\mathrm{d}t$. Let $N_m^k=(I_m^k{)}^{\ast }{I}_m^k$ be the normal operator of $I_m^k$. To what extent is a symmetric $m$-tensor field $f$ determined by the data $(N_m^{0}f,\dots ,N_m^{r}f)$ given for some $0\le r\le m$? The Saint Venant operator $W_m^r$ is a linear differential operator of order $m-r$ with constant coefficients on the space of symmetric $m$-tensor fields. We derive an explicit formula expressing $W_m^{r}f$ in terms of $(N_m^{0}f,\dots ,N_m^{r}f)$. The tensor field $W_m^{r}f$ represents the full local information on $f$ that can be extracted from the data $(N_m^{0}f,\dots ,N_m^{r}f)$.