both $T$ and $T'$ are identity operators. Then  is trivially satisfied and, as a consequence, one gets $$\begin{aligned}
\label{E:equivalence-of-identities}
X\hookrightarrow Y\quad \Leftrightarrow \quad Y'\hookrightarrow X'\end{aligned}$$ for every pair of r.i. spaces $X,Y$, again with the same embedding constant, see [@BS Chapter 1, Proposition 2.10].

We will say that an r.i. space $Y$ over $(S,\nu )$ is a *range partner* for a given r.i. space $X$ over $(R,\mu )$ with respect to a sublinear operator $T$ if $T\colon X\to Y$. We say that $Y$ is the *optimal range partner* for $X$ if one has $Y\hookrightarrow Z$ for every range partner $Z$ for $X$ with respect to $T$. We analogously define a *domain partner* and the *optimal domain partner*, that is, the largest possible domain space.

Throughout the paper the convention that $\frac{1}{\infty }=0$, and $0\cdot \infty =0$ is used without further explicit reference. We write $A\approx B$ when the ratio $A/B$ is bounded from below and from above by positive constants independent of appropriate quantities appearing in expressions $A$ and $B$.

The Hardy-Littlewood maximal operator {#sec3}
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In this section, the relevant r.i. spaces are considered over $\mathbb{R}^{n}$ endowed with the $n$-dimensional Lebesgue measure. The Lebesgue measure of a measurable set 