Range conditions for a spherical mean transform and global extension of solutions of Darboux equation

The transform under study is defined by integration of functions over spheres centered on a sphere. Such transform is of interest due to its applications in analysis and (thermoacoustic) tomography. The range of this transform has been described recently. Besides natural smoothness and support conditions, two type of conditions were involved: orthogonality condition and, in even dimension, moment condition. However, the moment condition was later derived from the other ones, which implies that orthogonality condition completely characterizes the range. We present a direct proof of this fact by proving existence of global extension for solutions of a certain boundary value problem for Darboux equation, associated with the spherical means. This extendibility phenomenon seems to be of independent interest.