The present paper brings a sweeping generalization of very new results obtained on the spherical framework exchanging unit spheres by compact two-point homogeneous spaces. We first prove a convenient characterization of a K-functional on this framework given by the rate of approximation of mean operators. Later, we apply such result in order to show that an abstract Hölder condition or finite order of differentiability assumption on kernels generating positive integral operators implies a sharp polynomial decay rates for eigenvalues sequences of such operators.