Limiting embeddings in smoothness Morrey spaces, continuity envelopes and applications

In this paper, the authors prove some Franke–Jawerth embedding for the Besov-type spaces Bp,qs,τ(Rn) and the Triebel–Lizorkin-type spaces Fp,qs,τ(Rn). By using some limiting embedding properties of these spaces and the Besov–Morrey spaces Nu,p,qs(Rn), the continuity envelopes in Bp,qs,τ(Rn), Fp,qs,τ(Rn) and Nu,p,qs(Rn) are also worked out. As applications, the authors present some Hardy type inequalities in the scales of Bp,qs,τ(Rn), Fp,qs,τ(Rn) and Nu,p,qs(Rn), and also give the estimates for approximation numbers of the embeddings from Bp,qs,τ(Ω), Fp,qs,τ(Ω) and Nu,p,qs(Ω) into C(Ω)C(Ω), where ΩΩ denotes the unit ball in RnRn.