Range of the generalized Radon transform associated with partial differential operators (Q1915569)

The authors consider the two partial differential operators \[ D_1= {\partial^2 \over {\partial \theta^2}}+ 4\alpha \cot g\theta {\partial \over {\partial \theta}} \quad \text{and} \quad D_2= {\partial^2 \over {\partial y^2}}+ [2 (2\alpha+ 1) \coth 2y] {\partial \over {\partial y}} -{1 \over {ch^2 y}} D_1+ (2\alpha+ 1)^2, \] \[ \alpha\in \mathbb{R}, \quad \alpha\geq 0 \quad \text{and} \quad (y, \theta)\in ]0, +\infty [\times ]0, {\textstyle {\pi \over 2}} [. \] They define a generalized Radon transform \({}^t{\mathfrak R}_\alpha\) and its dual \({\mathfrak R}_\alpha\) associated with these operators and they characterize the range of the transform \({}^t{\mathfrak R}_\alpha\).