On a Tauberian theorem with the remainder term and its application to the Weyl law

The purpose of this paper is twofold. First, we prove a generalization of the classical Tauberian theorem for the Laplace transform obtained by A. M. Subhankulov which gives an optimal bound for the remainder term. Second, we apply the Subhankulov theorem to a suitably transformed trace formula in the setting of symmetric spaces of real rank one and obtain an improved bound for the remainder term in the Weyl law. Our analysis is valid assuming an order of growth of the logarithmic derivative of the scattering determinant along imaginary axes.