Estimates for the best approximations of functions from the Nikol'skii-Besov class in the Lorentz space by trigonometric polynomials

We consider spaces of periodic functions of many variables, specifically, the Lorentz space $L_{p, \tau}(\mathbb{T}^{m})$ and the Nikol'skii–Besov space $S_{p, \tau, \theta}^{\bar{r}}B$, and study the best approximation of a function $f \in L_{p, \tau}(\mathbb{T}^{m})$ by trigonometric polynomials with the numbers of harmonics from a step hyperbolic cross. Sufficient conditions are established for a function $f \in L_{p, \tau_{1}}(\mathbb{T}^{m})$ to belong to a space $L_{q, \tau_{2}}(\mathbb{T}^{m})$ in the cases $1 <p <q <\infty$, $1 <\tau_{1}, \tau_{2} <\infty$ and $p = q$, $ 1 <\tau_{2} <\tau_{1} <\infty$. Estimates for the best approximations of functions from the Nikol'skii–Besov class $S_{p, \tau_{1}, \theta}^{\bar{r}}B$ in the norm of the space $L_{q, \tau_{2}}(\mathbb{T}^{m})$ are derived for different relations between the parameters $p$, $q$, $\tau_{1}$, $\tau_{2}$, and $\theta$. For some relations between these parameters, it is shown that the estimates are exact.