On Sobolev and Franke–Jawerth embeddings of smoothness Morrey spaces

Abstract

We study embeddings of Triebel–Lizorkin–Morrey spaces ${\mathcal {E}}^{s}_{p,u,q}({\mathbb {R}}^{d})$ within that scale as well as to classical spaces like $C({\mathbb {R}}^{d})$ or $L_r({\mathbb {R}}^{d})$. Here we obtain necessary and sufficient conditions for the continuity of it. Similarly we can deal with the situation when ${\mathbb {R}}^{d}$ is replaced by a bounded smooth domain $\Omega \subset {\mathbb {R}}^{d}$, now focussing, in addition, on compactness criteria. The second goal are embeddings of so-called Franke–Jawerth type, that is, $\mathcal{N}^{s_1}_{p_1,u_1,q_1}({\mathbb {R}}^{d}) \hookrightarrow {\mathcal {E}}^{s}_{p,u,q}({\mathbb {R}}^{d}) \hookrightarrow \mathcal{N}^{s_2}_{p_2,u_2,q_2}({\mathbb {R}}^{d})$, where the differential dimension is fixed, $s_1 - \frac{d}{p_1} = s_2 - \frac{d}{p_2}=s-\frac{d}{p}$, and $s_1 > s>s_2$.