Embeddings of generalised Morrey smoothness spaces

Abstract

We study embeddings between generalised Triebel-Lizorkin-Morrey spaces ${\mathcal{E}}_{\phi ,p,q}^s({\mathbb{R}}^d)$ and within the scales of further generalised Morrey smoothness spaces like ${\mathcal{N}}_{\phi ,p,q}^s({\mathbb{R}}^d)$, $B_{p,q}^{s,\phi }({\mathbb{R}}^d)$ and $F_{p,q}^{s,\phi }({\mathbb{R}}^d)$. The latter have been investigated in a recent paper by the first two authors (2023), while the embeddings of the scale ${\mathcal{N}}_{\phi ,p,q}^s({\mathbb{R}}^d)$ were mainly obtained in a paper of the first and last two authors (2022). Now we concentrate on the characterisation of the spaces ${\mathcal{E}}_{\phi ,p,q}^s({\mathbb{R}}^d)$. Our approach requires a wavelet characterisation of those spaces which we establish for the system of Daubechies' wavelets. Then we prove necessary and sufficient conditions for the embedding ${\mathcal{E}}_{{\phi }_1,p_1,q_1}^{s_1}({\mathbb{R}}^d)\hookrightarrow {\mathcal{E}}_{{\phi }_2,p_2,q_2}^{s_2}({\mathbb{R}}^d)$. We can also provide some almost final answer to the question when ${\mathcal{E}}_{\phi ,p,q}^s({\mathbb{R}}^d)$ is embedded into $C({\mathbb{R}}^d)$, complementing our recent findings in case of ${\mathcal{N}}_{\phi ,p,q}^s({\mathbb{R}}^d)$.