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This paper deals with Besov spaces of logarithmic smoothness $B_{p,r}^{0,b}$ formed by periodic functions. We study embeddings of $B_{p,r}^{0,b}$ into Lorentz-Zygmund spaces $L_{p,q}{\left(logL\right)}_{\beta }$. Our techniques rely on the approximation structure of $B_{p,r}^{0,b}$, Nikol’skiĭ type inequalities, extrapolation properties of $L_{p,q}{\left(logL\right)}_{\beta }$ and interpolation.

We investigate compact operators between approximation spaces, paying special attention to the limit case. Applications are given to embeddings between Besov spaces.