On Equivalent Normings in the Spaces $H_p^r$ on Compact Homogeneous Spaces

Let $M$ be an arbitrary compact Riemannian symmetric space of rank 1. The function spaces $H_p^r$ of Nikol'skii type were introduced earlier by means of averaged differences along geodesics. In the present paper we give an equivalent description of these spaces and norms in them by using the Laplace–Beltrami operator. The results obtained generalize the results of Nikol'skii and Lizorkin on the spaces $H_p^r$ over the sphere.