On a class of interpolation inequalities on the 2D sphere

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 for the embedding $H^1\hookrightarrow L^q$, $q<\infty$, are obtained in the Gagliardo-Nirenberg interpolation inequalities.
Bibliography: 25 titles.