We study embeddings between generalised Triebel–Lizorkin–Morrey spaces \(\cal{E}_{\varphi,p,q}^{s}(\mathbb{R}^{d})\) and within the scales of further generalised Morrey smoothness spaces like \(\cal{N}_{\varphi,p,q}^{s}(\mathbb{R}^{d})\) , B p,q s,φ (ℝd) and F p,q s,φ (ℝd). The latter have been investigated in a recent paper by the first two authors (2023), while the embeddings of the scale \(\cal{N}_{\varphi,p,q}^{s}(\mathbb{R}^{d})\) were mainly obtained in a paper of the first and last two authors (2022). Now we concentrate on the characterisation of the spaces \(\cal{E}_{\varphi,p,q}^{s}(\mathbb{R}^{d})\) . 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 \(\cal{E}_{\varphi_{1},p_{1},q_{1}}^{s_{1}}(\mathbb{R}^{d})\hookrightarrow\cal{E}_{\varphi_{2},p_{2},q_{2}}^{s_{2}}(\mathbb{R}^{d})\) . We can also provide some almost final answer to the question when \(\cal{E}_{\varphi,p,q}^{s}(\mathbb{R}^{d})\) is embedded into C(ℝd), complementing our recent findings in case of \(\cal{N}_{\varphi,p,q}^{s}(\mathbb{R}^{d})\) .
