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Decay rates for the sequence of eigenvalues of positive and compact integraloperators has been largely investigated for a long time in the literature. Inthis paper, the focus will be on positive integral operators acting on squareintegrable functions on the unit sphere and generated by a kernel satisfying aH\"older type assumption defined via average operators. In the approach to bepresented here, the decay rates will be reached from convenient estimations onthe eigenvalues of the operator themselves, with the help of specificproperties of a generic approximation operator defined through the so-calledgeneralized Jackson kernels. The decay rates have the same structure of thoseknown to hold in the cases in which the H\"older condition is the classicalone. Therefore, within the spherical setting, the abstract approach to beintroduced here extends some classical results on the topic.

Jackson kernels: a tool for analyzing the decay of eigenvalue sequences of integral operators on the sphere