A local uniqueness theorem for weighted Radon transforms (Q618994)

The weighted Radon transform in the Euclidean plane can be written as \[ R_{\rho}f(\omega,p)=\int_{\omega\cdot x=p}f(x)\rho(x,\omega)\,ds, \] where \(ds\) is the arc length measure on the line \(\omega\cdot x=\omega_{1} x_{1}+\omega_{2}x_{2}=p\), \(\omega(\omega_{1},\omega_{2})\in S^{1}\) is a unit vector, and \(\rho(x,\omega)\) is a weight function defined on \(\mathbb{R}^{2}\times S^{1}\). Another version of a weighted Radon transform in the plane may be given by \[ R_{m}f\left( \xi,\eta\right) =\int_{\mathbb{R}}f\left( x,\xi x+\eta\right) m\left( x,\xi,\eta\right)\, dx, \] where \(m\left( x,\xi,\eta\right) \) is a positive smooth function. \textit{S. Gindikin} [Inverse Probl. Imaging 4, No.~4, 649--653 (2010; Zbl 1206.44002)] proved the following uniqueness theorem for the weighted Radon transform. Assume that \(f\left( x,y\right)\) is a continuous function that vanishes for \(y<0\) and for \(\left| x\right| >A\) and that \[ R_{m}f\left( \xi,\eta\right) =0 \] for \(\left( \xi,\eta\right) \) in some neighborhood of the origin. Assume that \(m\left( x,\xi,\eta\right) \) is strictly positive and satisfies the condition \[ m_{\xi}^{\prime}\left( x,\xi,\eta\right) -xm_{\eta}^{\prime}\left( x,\xi,\eta\right) =(x a\left( \xi,\eta\right) +b\left( \xi,\eta\right) )m\left( x,\xi,\eta\right) , \tag{\(*\)} \] for some functions \(a\left( \xi,\eta\right) \) and \(b\left( \xi,\eta\right) \) that are independent of \(x\). Then \(f=0\) in some neighborhood of the \(x\)-axis. Consequently, if \(f\) has compact support and satisfies \(R_{f}f\left( \xi,\eta\right) =0\) for \(\xi\) in some neighborhood of the origin and all\(\;\eta\), where \(m\left( x,\xi,\eta\right) \) is positive and satisfies (\(*\)), then \(f=0.\). Inspired by Gindikin's result, the author of the paper under review proves the following uniqueness theorem: Assume that \(f\) is continuous, compactly supported, and satisfies \(R_{m}f\left( \xi,\eta\right) =0\) for an open, connected and unbounded set \(E\) of lines \(y=\xi x+\eta,\) and that \(m\left( x,\xi,\eta\right) \) is positive and satisfies (\(*\)), then \(f=0\) on the union of all lines in \(E\).