Microlocal analaysis of the Doppler transform on \(\mathbb{R}^3\)
The author considers the Doppler transform of a vector field \(F= (f_1,f_2,f_3)\) on \(\mathbb{R}^3\), defined by \[ DF(x, p)= \sum_{j= 1,2,3} \int_{\mathbb{R}} p_jf_j(x+ tp)\,dt, \] where \(x\in\mathbb{R}^3\) and \(p\in\mathbb{R}^3\) with \(|p|= 1\). The problem under consideration is the reconstruction of curl \(F\) from \(DF\), known only for a small subset of lines in \(\mathbb{R}^3\). The author gives an answer in terms of microlocal techniques, by finding a parametrix of the problem, in the case of \(DF\)-restricted to all lines passing through a fixed smooth curve. Namely, \(D\) is regarded in the frame of the Fourier integral operators with singular symbols introduced by \textit{V . Guillemin} and \textit{G . Uhlmann} [Duke Math. J. 48, 251--267 (1981; Zbl 0462.58030)]. The corresponding symbol is proved to be elliptic (modulo the symbol of the curl) and the parametrix is then constructed by symbolic calculus.
Fourier integral operators with singular symbols
scientific article; zbMATH DE number 2105067