the other hand, surprisingly little attention has been paid to the *sharpness* of the results, perhaps with an exception of results in different direction on optimality obtained e.g. in [@DS:07; @ST:16] and the references therein, where operators related to the Hardy averaging operator are studied, see also [@CR:02]. Optimal range spaces for Calderón operators are studied in the recent paper [@STZ:19].

In this paper we study the behaviour of classical operators on r.i. spaces, a class of function spaces that includes for example all Lebesgue, Lorentz, Orlicz, Lorentz-Zygmund spaces and more. Our focus is mainly on the optimality of function spaces.

We use the Hardy-Littlewood maximal operator $M$ to illustrate the results obtained and serve as an appetiser for the forthcoming attractions. Let $X$ be an r.i. space over $\mathbb{R}^{n}$ with associate space $X^{\prime }$ denote by $X^{\prime }(0,\infty )$ the representation space of $X^{\prime }$ and suppose that the function $\psi $ given by $\psi (t)=\chi _{(0,1)}(t)
\log (1/t)$ belongs to $X^{\prime }(0,\infty )$. Let $Y^{\prime }$ be the set of all $f$ such that $$\begin{aligned}
\varrho (f)=\left \Vert \int \nolimits _{t}^{\infty }f^{\ast }(s)s^{-1}
\,\mathrm{d}s\right \Vert _{X^{\prime }(0,\infty )}<\infty .\end{aligned}$$ Endowed with the norm $\varrho $, $Y^{\prime }$ is 