Besov and Triebel-Lizorkin spaces built over Morrey spaces provide a natural extension of the classical smoothness scales and have found important applications in the analysis of partial differential equations. More recently, these constructions have been generalised by replacing the Morrey parameter with an admissible function \( \varphi \), leading to the spaces

\( \mathcal{N}{\varphi,p,q}^s(\mathbb{R}^d), \qquad \mathcal{E}{\varphi,p,q}^s(\mathbb{R}^d), \qquad B_{p,q}^{s,\varphi}(\mathbb{R}^d), \qquad F_{p,q}^{s,\varphi}(\mathbb{R}^d). \)

These frameworks encompass both classical and Morrey-type spaces as special cases and allow for a finer description of local regularity. In this talk, we survey recent developments on these generalised smoothness spaces, focusing on embedding theorems and on how the behaviour of \( \varphi \) influences the structure and properties of the corresponding function spaces.

This talk is based on joint work with Dorothee Haroske (Friedrich Schiller University Jena), Leszek Skrzypczak (Adam Mickiewicz University, Poznań), and Zhen Liu (Beijing Normal University).