Dorothee D. Haroske, Zhen Liu, Susana D. Moura, Leszek Skrzypczak
We study embeddings between generalised Triebel-Lizorkin-Morrey spaces $\mathcal{E}_{\varphi, p, q}^s\left(\mathbb{R}^d\right)$ and within the scales of further generalised Morrey smoothness spaces like $\mathcal{N}_{\varphi, p, q}^s\left(\mathbb{R}^d\right)$, $B_{p, q}^{s, \varphi}\left(\mathbb{R}^d\right)$ and $F_{p, q}^{s, \varphi}\left(\mathbb{R}^d\right)$. The latter have been investigated in a recent paper by the first two authors (2023), while the embeddings of the scale $\mathcal{N}_{\varphi, p, q}^s\left(\mathbb{R}^d\right)$ were mainly obtained in a paper of the first and last two authors (2022). Now we concentrate on the characterisation of the spaces $\mathcal{E}_{\varphi, p, q}^s\left(\mathbb{R}^d\right)$. 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 $\mathcal{E}_{\varphi_1, p_1, q_1}^{s_1}\left(\mathbb{R}^d\right) \hookrightarrow \mathcal{E}_{\varphi_2, p_2, q_2}^{s_2}\left(\mathbb{R}^d\right)$. We can also provide some almost final answer to the question when $\mathcal{E}_{\varphi, p, q}^s\left(\mathbb{R}^d\right)$ is embedded into $C\left(\mathbb{R}^d\right)$, complementing our recent findings in case of $\mathcal{N}_{\varphi, p, q}^s\left(\mathbb{R}^d\right)$.
