Embeddings of block-radial functions — approximation properties and nuclearity

Let $R_{\gamma }{B}_{p,q}^s\left(R^d\right)$ be a subspace of the Besov space $B_{p,q}^s\left(R^d\right)$ that consists of block-radial (multi-radial) functions. We study the asymptotic behaviour of approximation numbers of compact embeddings $\mathrm{id}:R_{\gamma }{B}_{p_1,q_1}^{s_1}\left(R^d\right)\to R_{\gamma }{B}_{p_2,q_2}^{s_2}\left(R^d\right)$. Moreover, we find a sufficient and necessary condition for nuclearity of the above embeddings. Analogous results are proved for fractional Sobolev spaces $R_{\gamma }{H}_p^s\left(R^d\right)$.