By using the language of diffusion semigroups, it is possible to define and study classical operators in harmonic analysis. We introduce and develop this idea, and show two applications. First, we investigate fractional integrals and Riesz transforms in compact Riemannian spaces of rank one. Secondly, we carry out the study of operators associated with a discrete Laplacian, namely the fractional Laplacian, maximal heat and Poisson semigroups, square functions, Riesz transforms and conjugate harmonic functions.