involving capacitary estimates [@Mabook pp. 105 and 109]), Hansson [@Ha] and Brézis–Wainger [@BW] who used it to describe the sharp target space in a limiting Sobolev embedding (the spaces can be also traced in the works of Brudnyi [@B] and, in a more general setting, Cwikel and Pustylnik [@CP]). One of the benefits of using broken logarithmic functions consists in the fact that the underlying measure space can be considered to have either finite or infinite measure. For the detailed study of generalized Lorentz–Zygmund spaces we refer the reader to [@EOP; @EOP-broken; @OP; @FS].

We further define the spaces $L^{(p,q;{\mathbb{A}})}$ through the functionals $\varrho _{(p,q;{\mathbb{A}})}$ given 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, in an analogous way, all the other spaces involving various levels of logarithms.

Let $X$ and $Y$ be r.i. spaces over possibly different measure spaces $(R,\mu )$ and $(S,\nu )$, respectively, and let $T$ be an operator defined on $X$ with values in $\mathcal{M}(S,
\nu )$. We say that $T$ is *bounded* from $X$ to $Y$, a fact which is denoted by $T\colon X\to Y$, if there exists a positive constant $C$ such that $$\begin{aligned}
\|Tf\|_{Y}\leq 