Best constants in a class of polymultiplicative inequalities for derivatives

Best constants are found in a class of multiplicative inequalities with $k$ factors that give an estimate of the $C$-norm of a function (in $\mathbb R^n$ or on $\mathbb S^n$) in terms of the product of the $L_2$-norms of fractional powers of the Laplace operator. Special attention is given to the detection of the cases of equality of the corresponding constants on the sphere and in Euclidean space.