A generalized Helmholtz-type decomposition of symmetric tensor fields and applications to ray transforms

We study a solenoidal-potential type decomposition of a symmetric m-tensor field in \Rb^2, and its implications to injectivity questions for the momentum and elastic ray transforms. For symmetric tensor fields, a general decomposition with a restriction on the dimension and order of the decomposition was proved in~\cite{Rohit_Suman}. We extend the result to dimension 2 under a mean-zero assumption. We use the decomposition in 2 dimensions to prove the injectivity of the momentum and elastic ray transforms. We also prove a connection between the two integral transforms for 2-tensors.