Semiclassical estimates for eigenvalue means of Laplacians on spheres

We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper and lower bounds involving asymptotically sharp shift terms, and we extend them to domains of $\mathbb S^d$. We also prove a Berezin-Li-Yau inequality for domains contained in the hemisphere $\mathbb S^2_+$. Moreover, we consider polyharmonic operators for which we prove analogous results that highlight the role of dimension for P\'olya-type inequalities. Finally, we provide sum rules for Laplacian eigenvalues on spheres and compact two-point homogeneous spaces.