Microlocal Analysis of the Doppler Transform on R³

Abstract

The Doppler transform of a vector field $F = (f_1,f_2,f_3)$ on $\mathbb{R}^3$ is defined by \[\displaystyle\mathcal{D}F(x,\omega) = \sum_j\int_\mathbb{R} \omega_j f_j(x+t\omega)\, dt~,\] where $x\in \mathbb{R}^3$ and $\omega \in S^2$ specifies the direction of a line passing through $x$. In practical applications, $\mathcal{D}F$ is known only for a small subset of lines in $\mathbb{R}^3$. In this article, we deal with the case of $\mathcal{D}F$ restricted to all lines passing through a fixed smooth curve. Using techniques from microlocal analysis, we study the problem of recovering the wavefront set of $\mbox{curl}(F)$ from that of the restricted Doppler transform of $F$.