Improved estimates for the linear Molodensky problem

The paper deals with the linearized Molodensky problem, when data are supposed to be square integrable on the telluroid S, proving that a solution exists, is unique and is stable in a space of harmonic functions with square integrable gradient on S. A similar theorem has already been proved by Sansò and Venuti (J Geod 82:909–916, 2008). Yet the result basically requires that S should have an inclination of less than \(60^\circ \) ${60}^{\circ }$ with respect to the vertical, or better to the radial direction. This constraint could result in a severe regularization for the telluroid specially in mountainous areas. The paper revises the result in an effort to improve the above estimates, essentially showing that the inclination of S could go up to \(75^\circ \) ${75}^{\circ }$ . At the same time, the proof is made precise mathematically and hopefully more readable in the geodetic community.