Berezin--Li--Yau inequalities on domains on the sphere

We prove Berezin–Li–Yau inequalities for the Dirichlet and Neumann eigenvalues on domains on the sphere ${\mathbb{S}}^{d-1}$. The case of ${\mathbb{S}}^2$ is treated in greater detail, including the vector Dirichlet Laplacian and the Stokes operator.