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In this paper we prove Nikolskii's inequality on general compact Lie groupsand on compact homogeneous spaces with the constant interpreted in terms of theeigenvalue counting function of the Laplacian on the space, giving the bestconstant for certain indices, attained on the Dirichlet kernel. Consequently,we establish embedding theorems between Besov spaces on compact homogeneousspaces, as well as embeddings between Besov spaces and Wiener and Beurlingspaces. We also analyse Triebel-Lizorkin spaces and $\beta$-versions of Wienerand Beurling spaces and their embeddings, and interpolation properties of allthese spaces.

Nikolskii inequality and Besov, Triebel-Lizorkin, Wiener and Beurling spaces on compact homogeneous manifolds