  We prove estimates for the $L^p$-norms of systems of functions and divergence
free vector functions that are orthonormal in the Sobolev space $H^1$ on the 2D
sphere. As a corollary, order sharp constants in the embedding
$H^1\hookrightarrow L^q$, $q<\infty$, are obtained in the Gagliardo--Nirenberg
interpolation inequalities.
