the collection of all measurable functions $g$ on $Q$ satisfying $\int _{Q}|g(x)|(\log (1+|g(x)|)^{\alpha }\mathrm{d}x<\infty $, but we add the information that the range space cannot be improved in any way when the competing spaces are rearrangement invariant. Similar examples are even more interesting when the functions act on a set of unbounded measure, say, $\mathbb{R}^{n}$. We will for example prove that if $X$ is the space equipped with the norm $\|f\|_{X}=\int _{0}^{\infty }f^{*}(t)w(t)\,dt$, where $$\begin{aligned}
w(t)=(1-\log t)^{\alpha _{0}}\chi _{(0,1)}+(1+\log t)^{\alpha _{\infty }}
\chi _{[1,\infty )}\end{aligned}$$ and $\alpha _{0}\geq 1$ and $\alpha _{\infty }\in [-1,0]$, then the optimal (smallest possible) r.i. range space $Y$ such that $$\begin{aligned}
M\colon X(\mathbb{R}^{n})\to Y(\mathbb{R}^{n})\end{aligned}$$ is the space whose associate space has norm $$\begin{aligned}
\|f\|=\sup _{0<t<\infty }w(t)\sp{-1}\int _{t}^{\infty }f^{*}(s)\,\frac{
\mathrm{d}s}{s},\quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ Such results have not been available before, and the latter norm cannot be identified with any customary known one.

We get analogous sets of examples for other operators, too. For example in the case of the fractional maximal operator we essentially improve some results from earlier papers such as [@EO; @EOP; @EOP-broken; @OP].

Preliminaries {#sec2}
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In this section we collect all the background material that will be used in the paper. We start 