  The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$
of periodic functions of many variables and $S_{p,\tau,\theta}^{0,
\overline{b}}\mathbf{B}$, $S_{p, \tau, \theta}^{0, \overline{b}}B$ -- spaces of
functions with mixed logarithmic smoothness. The article establishes necessary
and sufficient conditions for embedding the spaces $S_{p, \tau, \theta}^{0,
\overline{b}}\mathbf{B}$ and $S_{p, \tau, \theta}^{ 0, \overline{b}}B$ into
each other.
