the functionals $
\varrho _{p,q;{\mathbb{A}}}$ and $\varrho _{p,q;{\mathbb{A}},{\mathbb{B}}}$ on $\mathcal{M}_{+}(R,
\mu )$ by $$\begin{aligned}
\varrho _{p,q;{\mathbb{A}}}(f)=
\left \|  t^{\frac{1}{p}-\frac{1}{q}}
\ell ^{{\mathbb{A}}}(t) f^{*}(t)\right \|  _{L^{q}(0,\infty )}\end{aligned}$$ and $$\begin{aligned}
\varrho _{p,q;{\mathbb{A}},{\mathbb{B}}}(f)=
\left \|  t^{\frac{1}{p}-
\frac{1}{q}}\ell ^{{\mathbb{A}}}(t)\ell \ell ^{{\mathbb{B}}}(t)
f^{*}(t)\right \|  _{L^{q}(0,\infty )},\end{aligned}$$ where $$\begin{aligned}
\ell ^{{\mathbb{A}}}(t)=
{\begin{cases}
(1-\log t)^{\alpha _{0}}
&\textup{if}\ t\in (0,1),
\cr
(1+\log t)^{\alpha _{\infty }}
&\textup{if}\ t\in [1,\infty )
\end{cases}}
$$ and $$\begin{aligned}
\ell \ell ^{{\mathbb{B}}}(t)=
{\begin{cases}
(1+\log (1-\log t))^{\beta _{0}}
&\textup{if}\ t\in (0,1),
\cr
(1+\log (1+\log t))^{\beta _{\infty }}
&\textup{if}\ t\in [1,\infty ).
\end{cases}}
$$ The set $L^{p,q;{\mathbb{A}}}$, defined as the collection of all $f\in \mathcal{M}(R,\mu )$ satisfying $\varrho _{p,q;{\mathbb{A}}}(|f|)<
\infty $, is called a *Lorentz–Zygmund space*, and the set $L^{p,q;{\mathbb{A}},{\mathbb{B}}}$, defined as the collection of all $f\in \mathcal{M}_{+}(R,\mu )$ satisfying $
\varrho _{p,q;{\mathbb{A}},{\mathbb{B}}}(|f|)<\infty $, is called a *generalized Lorentz–Zygmund space*. The functions of the form $\ell ^{{\mathbb{A}}}$, $\ell \ell ^{{\mathbb{B}}}$ are called *broken logarithmic functions*. The spaces of this type proved to be quite useful since they provide a common roof for many customary spaces. These include not only Lebesgue spaces and Lorentz spaces, but also all types of exponential and logarithmic Zygmund classes, and also the spaces discovered independently by Maz’ya (in a somewhat implicit form 