About the unimprobality of the limiting embedding theorem for different metrics in the Lorentz spaces with Hermite's weight

In this article we obtained inequality of different metrics in the Lorentz spaces with Hermit's weight for multiple algebraic polynomials. On this basis we established a sufficient condition of embedding of different metrics in the Lorenz spaces with Hermite's weight. Its unimprobality is shown in terms of the “extreme function”.
Let $f\in L_{p,\theta}(\mathbb R_n;\rho_n)$, $1\leq p<+\infty$, $1\leq\theta\leq+\infty$. The sequense $t\{l_k\}_{k=0}^{+\infty}\subset\mathbb N$ is such that $l_0=1$ and $l_{k+1}\cdot l_k^{-1}>a_0>1$, $\forall k\in\mathbb Z^+$. $f(\bar x)=\sum_{k=0}^{+\infty}\Delta_{l_k,\dots,l_k}(f;\bar x)$ is some presentation of the functions in the metric $L_{p,\theta}(\mathbb R_n;\rho_n)$, where $\Delta_{l_0,\dots,l_0}(f;\bar x)=T_{1,\dots,1},\Delta_{l_k,\dots,l_k}(f;\bar x)=T_{l_k,\dots,l_k}(\bar x)-T_{l_{k-1},\dots,l_{k-1}}(\bar x)$, $\forall k\in\mathbb N$. Here
$$ T_{l_k,\dots,l_k}(\bar x)=\sum_{m_1=0}^{l_k-1}\dots\sum_{m_n=0}^{l_k-1}a_{m_1,\dots,m_n}\prod^n_{i=1}x^{m_i}_i $$
are algebraic polynomials for all $k\in\mathbb Z^+$.
$1^0$. If the series
$$ A(f)_{p\theta}=\sum_{k=0}^{+\infty}l_k^{\tau(\frac n{2p}-\frac n{2q})}\|\Delta_{l_k,\dots,l_k}(f)\|_{L_{p,\theta}(\mathbb R_n;\rho_n)}^\tau $$
converge under some $q$ and $\tau$: $p<q<+\infty$, $0<\tau<+\infty$, then $f\in L_{q,\tau}(\mathbb R_n;\rho_n)$ and we have the inequality
$$ \|f\|_{L_{q,\tau}(\mathbb R_n;\rho_n)}\leq C_{pq\theta\tau n}\times(A(f)_{p\theta})^\frac1\tau. $$

$2^0$. The condition $1^0$ is unimprovable in the sense that there exists a function $f_0\in L_{p,\theta}(\mathbb R_n;\rho_n)$ and $A(f_0)_{p\theta}$ diverges for it and $f_0\notin L_{q,\tau}(\mathbb R_n;\rho_n)$. At the same time, the function $f_0\in L_{q-\varepsilon,\tau}(\mathbb R_n;\rho_n)$ for all $\varepsilon>0$: $p<(q-\varepsilon)<q$.