Embedding theorems for symmetric spaces of measurable functions

Let $m$ be the usual Lebesgue measure on $\mathbb{R}_+ = [0,+\infty)$. Dealing with symmetric (rearrangement invariant) spaces $\mathbf{E}$ on the standard measure space $(\mathbb{R}_+,m)$, we treat the following embeddings:
$$
\mathbf{L}_1\cap\mathbf{L}_\infty \subseteq \mathbf{\Lambda}^0_{\widetilde{V}}\subseteq \mathbf{E}^0\subseteq \mathbf{E}\subseteq \mathbf{E}^{11}\subseteq \mathbf{M}_{V_*} \subseteq \mathbf{L}_1+\mathbf{L}_\infty  ,
$$
where $\mathbf{E}^0= cl_\mathbf{E}(\mathbf{L}_1\cap\mathbf{L}_\infty)$ – is the closure of $\mathbf{L}_1\cap\mathbf{L}_\infty$ в $\mathbf{E}$, $\mathbf{E}^{11}=(\mathbf{E}^1)^1$ is the second associate space of $\mathbf{E}$, $V(x)= \|1_{[0,x]}\|_\mathbf{E}$ is the fundamental function of the symmetric space $\mathbf{E}$, $\displaystyle{V_*(x)= \frac{x}{V(x)}1_{(0,\infty)}(x)}$, $\widetilde{V}$ is the least concave majorant of $V$, $\mathbf{\Lambda}_{\widetilde{V}} $ and $ \mathbf{M}_{V_*}$ are the Lorentz and Marcinkiewicz spaces with the weights $\widetilde{V}$ and $V_*$ respectively, $\mathbf{\Lambda}^0_{\widetilde{V}}=cl_{\mathbf{\Lambda}_{\widetilde{V}}}(\mathbf{L}_1\cap\mathbf{L}_\infty) $.
The space $\mathbf{\Lambda}^0_{\widetilde{V}}$ is the minimal part of the Lorentz space $\mathbf{\Lambda}_{\widetilde{V}}.$ It is the smallest symmetric space on $\mathbb{R}_+$ whose fundamental function $\phi_{\mathbf{\Lambda}^0_{\widetilde{V}}} = \widetilde{V}$ is equivalent to $V.$ The Marcinkiewicz space
$\mathbf{M}_{V_*}$ is largest symmetric space on $\mathbb{R}_+$ satisfying $\phi\mathbf{M}_{V_*} = \phi\mathbf{E} = V.$
The inclusion $\mathbf{\Lambda}_{\widetilde{V}}\subseteq \mathbf{E}$ claimed in [3, II.5.4, Th. 5.5] fails in general. Although, it is true, for example, if $V(+\infty) = \infty$ (the space $\mathbf{\Lambda}_{\widetilde{V}}$ is minimal), or if the space $\mathbf{E}$ itself is maximal $(\mathbf{E} = \mathbf{E}_{11}).$
The embeddings and natural inequalities for corresponding norms are studied in detail.
A full English version of the work will be published in [7].

Keywords: Symmetric spaces, Lorentz and Marcinkiewicz spaces, embedding theorems.