Rotation invariant subspaces of Besov and Triebel-Lizorkin space: compactness of embeddings, smoothness and decay of functions

Abstract

Let $H$ be a closed subgroup of the group of rotation of $\mathbb{R}^n$. The subspaces of distributions of Besov-Lizorkin-Triebel type invariant with respect to natural action of $H$ are investigated. We give sufficient and necessary conditions for the compactness of the Sobolev-type embeddings. It is also proved that $H$-invariance of function implies its decay properties at infinity as well as the better local smoothness. This extends the classical Strauss lemma. The main tool in our investigations is an adapted atomic decomposition.