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The transform under study is defined by integration of functions over spherescentered on a sphere. Such transform is of interest due to its applications inanalysis and (thermoacoustic) tomography. The range of this transform has beendescribed recently. Besides natural smoothness and support conditions, two typeof conditions were involved: orthogonality condition and, in even dimension,moment condition. However, the moment condition was later derived from theother ones, which implies that orthogonality condition completely characterizesthe range. We present a direct proof of this fact by proving existence ofglobal extension for solutions of a certain boundary value problem for Darbouxequation, associated with the spherical means. This extendibility phenomenonseems to be of independent interest.

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