measurable $W$-invariant functions $f:{\mathfrak{a}}\to{\mathbb{C}}$ such that $$\|f\|_{(q),\psi}:=\Bigl(\int_{{\mathfrak{a}}_+}\vert f(x)\vert^q\psi(x)^{q-2}\,d\mu(x)\Bigr)^{1/q}<\infty.$$ In other words, $f$ belongs to $L^{(q)}_\psi$ if and only if $f\cdot\psi^{1-\frac{2}{q}}$ belongs to $L^q$.

The next result is a generalization of theorem \[thm.1\] in the introduction.

\[thm.HY-qlarge\] Let $q>2$ and $f\in L^{(q)}_{\psi_h}({\mathfrak{a}}_+)$. There exists a positive constant $D_q$ independent of $f$ such that $$\frac{1}{|W|}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^q\,d\nu(\lambda)\leq D_q^q\|f\|^q_{(q),\psi}.$$

Let $f$ be a simple function on $A$ and let $Tf(\lambda)=\mathcal{F}f(\lambda)$ (we do not need to add weights to the operator that enters the interpolation argument). Then $\|Tf\|^*_{\infty,\infty}=\|Tf\|_\infty\leq C\|f\|_1=\|f\|^*_{1,1}$, and by the Plancherel theorem it furthermore holds that $\|Tf\|_{2,\infty}^*\leq\|Tf\|^*_{2,2}\leq\|f\|_2\leq\|f\|^*_{2,1}$. By interpolation (cf. theorem \[thm.lorentz-inter\]) it follows that $\|Tf\|^*_{q,q}\leq \|f\|^*_{q',q}$.

Now define $g(x)=f(x)\psi_h(x)^{1-\frac{2}{q}}$, where $\psi_h(x)=\cosh(\|x\|)J(x)$. Then $g$ belongs to $L^q(A,d\mu)^W$ by hypothesis, since $$\|g\|^q_q=\int_A\vert f(x)\vert^q\vert\psi_h(x)\vert^{q-2}\,d\mu(x) = \|f\|^q_{(q),\psi_h}$$ It follows from the sublevel set estimate implied by $\psi_h$ being a Young function for $({\mathfrak{a}},\Sigma,m)$ that $$\mu\bigl(\{x\,:\,\vert\psi_h(x)\vert^{\frac{2}{q}-1}>t\}\bigr) = \mu\Bigl(\Bigl\{x\,:\,\vert\psi_h(x)\vert^{1-\frac{2}{q}}<\frac{1}{t}\Bigr\}\Bigr)\leq Ct^{-\frac{q}{q-2}},$$ whence $\psi^{\frac{2}{q}-1}$ belongs to $L^{r,\infty}(A,d\mu)$, where $r=\frac{q}{q-2}$. By an application of O’Neil’s theorem \[thm.Oneil\] it is seen that $$\begin{split}
\frac{1}{|W|}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^q\,d\nu(\lambda)&\leq \|f\|^*_{p,q}\leq \|g\|_q\|\psi_h\|^*_{t,\infty}\\ & \leq C\int_A|f(x)|^q|\psi_h(x)|^{q-2}J(x)\,dx = c\|f\|^p_{(q),\psi_h}
\end{split}$$ which was the desired conclusion for simple functions. The extension to general functions in $L^{(q)}_{\psi_h}({\mathfrak{a}}_+)$ now follows by standard density arguments.

As briefly mentioned in the introduction, Ray and Sarkar were 