On embedding of Besov spaces of zero smoothness into Lorentz spaces

We show that the zero smoothness Besov space $B_{p,q}^{0,1}$ does not embed into the Lorentz space $L_{p,q}$ unless $p=q$; here $p,q\in (1,\mathrm{\infty })$. This answers negatively a question proposed by O. V. Besov.