On a class of interpolation inequalities on the 2D sphere

We prove estimates for the $L^p$-norms of systems functions and divergence-free vector that are orthonormal in Sobolev space $H^1$ on 2D sphere. As a corollary, order sharp constants embedding $H^1\hookrightarrow L^q$, $q<\infty$, obtained Gagliardo-Nirenberg interpolation inequalities. Bibliography: 25 titles.