Some embeddings of Morrey spaces with critical smoothness

Abstract

We study embeddings of Besov-Morrey spaces ${\cal N}^{s}_{u,p,q}}({\mathbb R}^d)$ and of Triebel-Lizorkin-Morrey spaces ${\cal E}^{s}_{u,p,q}}({\mathbb R}^d)$ in the limiting cases when the smoothness $s$ equals $s_0=d\max (1/u-p/u,0)$ or $s_{\infty }=d/u$, which is related to the embeddings in $L_1^{loc}({\mathbb{R}}^d)$ or in $L_{\infty }({\mathbb{R}}^d)$, respectively. When $s=s_0$ we characterise the embeddings in $L_1^{loc}({\mathbb{R}}^d)$ and when $s=s_{\infty }$ we obtain embeddings into Orlicz-Morrey spaces of exponential type and into generalised Morrey spaces.