r.i. norm and  is satisfied, where $X=X(\varrho )$. Moreover, $X$ is the optimal largest rearrangement-invariant space for which  holds.

Conversely, if  is not true, then there does not exist an r.i. space $X$ for which  holds.

In our final result of this section we present a collection of nontrivial examples based on Lorentz–Zygmund spaces.

\[T:-maximal-operator-GLZ\] Let $p,q\in [1,\infty ]$, ${\mathbb{A}}\in \mathbb{R}^{2}$. Then $$\begin{aligned}
M\colon L^{p,q; {\mathbb{A}}} \to
\left\{
\begin{array}{l@{\quad }l@{\quad }l}
L^{1, 1, \mathbb{A} - 1},   & p = 1, q = 1, \alpha _{0}\geq 1,
\alpha _{\infty }< -1, &\mathrm{(a)}
\label{E:maximal_p1}
\\
Y,
& p = 1, q = 1, \alpha _{0}\geq 1, -1\leq \alpha _{\infty }
\leq 0, & \mathrm{(b)}
\label{E:maximal_p2}
\\
L^{p,q;{\mathbb{A}}}, & 1< p <\infty ~ or\\
& p=\infty , 1\leq q<\infty , \alpha _{0} + \frac{1}{q} < 0~ or \\
& p=\infty , q=\infty , \alpha _{0}\leq 0,
\end{array}\right.\end{aligned}$$ where $Y$ is the unique rearrangement-invariant space whose associate space $Y'$ satisfies $$\begin{aligned}
\|f\|_{Y'}=\sup _{0<t<\infty }\ell ^{-{\mathbb{A}}}(t)\int _{t}^{\infty
}f^{*}(s)\,\frac{{\mathrm{d}}s}{s}, \quad f\in \mathcal{M}_{+}(\mathbb{R}^{n}).\end{aligned}$$ These spaces are the optimal range partners with respect to $M$.

We note that the space $Y'$, given in terms of an operator-induced norm, cannot be expressed in terms of a Lorentz–Zygmund norm.

We 