Spaces of generalized smoothness in the critical case: Optimal embeddings, continuity envelopes and approximation numbers

We study necessary and sufficient conditions for embeddings of Besov spaces of generalized smoothness Bp,qσ,N(Rn) into generalized Hölder spaces Λ∞,rμ(⋅)(Rn) when s¯(Nτ−1)>0 and τ−1∈ℓq′, where τ=σN−n/p. A borderline situation, corresponding to the limiting situation in the classical case, is included and give new results. In particular, we characterize optimal embeddings for BB-spaces.As immediate applications of our results we obtain continuity envelopes and give upper and lower estimates for approximation numbers for the related embeddings.We also consider the analogous results for the Triebel–Lizorkin spaces of generalized smoothness Fp,qσ,N(Rn).