leave it to the interested reader to write down the details.

A *Young function* is a measurable function $\psi:{\mathfrak{a}}_+\to{\mathbb{R}}$ with the property that $\mu(\{x\in{\mathfrak{a}}\,:\,\vert\psi(x)\vert\leq t\})\lesssim t$ for all $t>0$.

\[example.young\] In ${\mathbb{R}}^n$, the function $\psi(x)=\|x\|^m$ is a Young function in ${\mathbb{R}}^n$ if and only if $m=n$, since $|\{x\in{\mathbb{R}}^n\,:\,\|x\|^m<t\}|=|B(0,t^{1/m})|=Ct^{n/m}$. Since norms on ${\mathbb{R}}^n$ are equivalent, the $n$.th power of *any* norm on ${\mathbb{R}}^n$ gives rise to a Young function.

It is easy to construct Young functions associated with $({\mathfrak{a}},\Sigma,m)$. An infinite family of examples is given by $\psi(x)=h(x)J(x)$, where $h(x)=h_0(\|x\|)$ is radial and satisfies $\int_0^\infty \frac{s^n-1}{h_0(s)}ds<\infty$. In this case, $$\label{ineq.sublevel}
\mu(\{x\,:\,\vert\psi(x)\vert\leq t\})\leq t\int_{{\mathfrak{a}}}\frac{dx}{h(x)} =Ct\int_0^\infty\frac{s^{n-1}}{h_0(s)}\,ds = C't$$ for every $t>0$. For many purposes the estimate in is too crude, however. The norm power $\|\cdot\|^n$ would not meet the requirement that $\int_0^\infty s^{n-1}/s\,ds$ be finite, for example, so the estimate is only sensible when the measure of the sublevel sets $\{x\,:\,|\psi(x)|<t\}$ cannot be estimated directly. One such example is the following.

The *hyperbolic* Young function associated with $({\mathfrak{a}},\Sigma,m)$ is the function $\psi_h(x)=\cosh(\|x\|)J(x)$. This is a root system analogue of the Young function considered in [@Mohanty-II Lemma 4.4].

Let $\psi$ be a Young function for $({\mathfrak{a}},\Sigma,m)$ and $q>2$. The space $L^{(q)}_\psi({\mathfrak{a}}_+)$ consists of all 