Function spaces of Nikolskii type on compact manifold

Bondioli, Cristiana

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 3 (1992) no. 3, pp. 185-194

Résumé

Nikolskii spaces were defined by way of translations on \( \mathbb{R}^{n} \) and by way of coordinate maps on a differentiable manifold. In this paper we prove that, for functions with compact support in \( \mathbb{R}^{n} \), we get an equivalent definition if we replace translations by all isometries of \( \mathbb{R}^{n} \). This result seems to justify a definition of Nikolskii type function spaces on riemannian manifolds by means of a transitive group of isometries (provided that one exists). By approximation theorems, we prove that - for homogeneous spaces of compact connected Lie groups - our definition is equivalent to the classical one.