Smoothness spaces related to Morrey spaces – a survey. I

Abstract

We discuss different strategies to introduce smoothness spaces related to Morrey spaces. In particular, we consider the Nikol’skij-Besov type spaces \( B^{s,\tau}_{p,q} \) and the Lizorkin-Triebel type spaces \( F^{s,\tau}_{p,q} \), and compare them with other possibilities of defining smoothness spaces of Nikol’skij–Besov–Lizorkin–Triebel type related to Morrey spaces. Altogether we discuss eight scales of function spaces: \( B^{s,\tau}_{p,q} \), \( F^{s,\tau}_{p,q} \), \( N^{s}_{p,q,u} \), \( E^{s}_{p,q,u} \), \( B^{s,\tau}_{p,q,\text{unif}} \), and \( F^{s,\tau}_{p,q,\text{unif}} \). Differences between the scales \( B^{s,\tau}_{p,q} \) and \( N^{s}_{p,q,u} \) (for \( q<\infty \)) are discussed in detail.