Integral operators on the sphere generated by positive definite smooth kernels

We consider integral operators on the unit sphere generated by positive definite kernels. Under smoothness conditions of Lipschitz-type on the kernel, we obtain a decay rate for the eigenvalues of the integral operator. The approach we have chosen is a multi-dimensional version, adapted to the spherical setting, of a known procedure used in the analysis of a similar problem for integral operators on the interval [0, 1]. In addition to spectral theory, the critical arguments in the paper involve the use of special covers of the sphere generated by quadrature formulas. The estimates themselves are comparable to others in the literature.