Eigenvalues of Integral Operators Defined by Smooth Positive Definite Kernels

We consider integral operators defined by positive definite kernels \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K : X \times X \rightarrow {\mathbb{C}}$$\end{document}, where X is a metric space endowed with a strictly-positive measure. We update upon connections between two concepts of positive definiteness and upgrade on results related to Mercer like kernels. Under smoothness assumptions on K, we present decay rates for the eigenvalues of the integral operator, employing adapted to our purposes multidimensional versions of known techniques used to analyze similar problems in the case where X is an interval. The results cover the case when X is a subset of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}^{m}$$\end{document} endowed with the induced Lebesgue measure and the case when X is a subset of the sphere Sm endowed with the induced surface Lebesgue measure.