We derive simple conditions, which are both necessary and sufficient, for the validity of sharp local embeddings $H^{\sigma,\alpha} X({\bf R}^{n}) \hookrightarrow Y({\bf {\Omega})$ and sharp global embeddings $H^{\sigma,\alpha}, X({\bf R}^{n}) \hookrightarrow Z({\bf R}^{n})$. Here $H^{\sigma,\alpha}$, stands for a Bessel potential operator involving the classical smoothness $\alpha$ and logarithmic smoothness $\alpha$, X, Y and Z are (generalized) Lorentz–Zygmund spaces, and $\Omega\subset {\bf R^{n}}$ is an open subset whose n-dimensional Lebesgue measure is finite. Our results extend those of [18] and improve them especially in the extreme cases when X is close to L1, or Y and Z are close to $L^{\infty}$, or when $\sigma=0$ and $\alpha > 0$.