Embeddings of Generalised Morrey Smoothness Spaces

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 ^s,φ_p,q (ℝ^d) and F ^s,φ_p,q (ℝ^d). 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(ℝ^d), complementing our recent findings in case of \(\cal{N}_{\varphi,p,q}^{s}(\mathbb{R}^{d})\).