Optimal local embeddings of Besov spaces involving only slowly varying smoothness

The aim of the paper is to establish (local) optimal embeddings of Besov spaces $B_{p,r}^{0,b}$ involving only a slowly varying smoothness $b$. In general, our target spaces are outside of the scale of Lorentz-Karamata spaces and are related to small Lebesgue spaces. In particular, we improve results from [CGO11b], where the targets are (local) Lorentz-Karamata spaces. To derive such results, we apply limiting real interpolation techniques and weighted Hardy-type inequalities.