2026/02/21 by Antti Kykkänen, Rohit Kumar Mishra, Suman Kumar Sahoo
#math.AP
We study a solenoidal-potential type decomposition of a symmetric m-tensor field in \Rb2, 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 [16]. 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. Later, we use our decomposition to prove the injectivity of integral transforms, including longitudinal and elastic ray transforms, without any mean-zero assumption on tensor fields. Next, we explore connections between different integral transforms and use these to relate their corresponding properties.