P\'{o}lya-type inequalities on spheres and hemispheres

Given an eigenvalue $\lambda$ of the Laplace-Beltrami operator on $n-$spheres and $-$hemi\-spheres, we characterise those with the lowest and highest orders which equal $\lambda$ and for which P\'{o}lya's conjecture holds and fails. We further derive P\'{o}lya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues. This allows us to measure the deviation from the leading term in the Weyl asymptotics for eigenvalues on spheres and hemispheres. As a direct consequence, we obtain similar results for domains which tile hemispheres.