1.  Let $1<q\leq 2$ be fixed. For $f\in L^p({\mathfrak{a}},d\mu)^W$ with $1<p\leq q$, $$\Bigl(\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda+\eta)|^r(\|\lambda\||{\mathbf{c}}(\lambda)|^{-2})^{r/p'-1}\,d\nu(\lambda)\Bigr)^{1/r}\leq C_{p,q,\eta}\|f\|_p$$ for every $\eta$ in the interior of $C(\epsilon_p\rho)$, where $\frac{1}{r}=1-\frac{q'-1}{p'}$.

2.  Let $2\leq q<\infty$ be fixed. For $f\in L^{(p)}({\mathfrak{a}})$ with $q\leq p<\infty$, $$\Bigl(\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda+\eta)|^p\,d\nu(\lambda)\Bigr)^{1/p}\leq C_{p,q,\eta}\|f\|_{(p)}$$ for every $\eta$ in the interior of $C(\epsilon_p\rho)$.

Note that the special case $p=q=r=2$ recovers the a special case of the Hausdorff–Young inequality in lemma \[HL-ineq2\] and the Plancherel formula as an inequality in the case $\eta=0$.

Fix $q\in(1,2]$ and consider the measure spaces $({\mathfrak{a}}_+,d\mu)$ and $(i{\mathfrak{a}}^+,d\overline{\nu}(\lambda)$, where $d\overline{\nu}(\lambda)=\|\lambda\|^{-nq}|{\mathbf{c}}(\lambda)|^{-2(1-nq)}\,d\lambda$. Define $Tf$, $f\in L^p({\mathfrak{a}},d\mu)^W$, by $Tf(\lambda)=|\mathcal{F}f(\lambda+\eta)|(\|\lambda\||{\mathbf{c}}(\lambda)|^{-2})^{\frac{q\cdot n}{q'}}$. It then follows from lemma \[lemma.NPP-2\] that $$\|Tf\|_{q'}^{q'} =\int_{i{\mathfrak{a}}^*} |Tf(\lambda)|^{q'}\|\lambda\|^{-nq}|{\mathbf{c}}(\lambda)|^{-2(1-nq)}\,d\lambda=
\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda+\eta)|^{q'}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda \leq C_{p,q,\eta}\|f\|_q^{q'},$$ so $T$ is of strong type $(q,q')$.

The operator $T$ is furthermore of weak type $(1,1)$, as we shall now prove. The argument is nearly the same as in the proof of theorem \[thm.HY-qlarge\] but relies on a clever trick employed in the proof of [@Ray-Sarkar-trans Theorem 4.11]. For $t>0$ define the set $E_t(f)=\{\lambda\in i{\mathfrak{a}}^*\,:\,|Tf(\lambda)>t\}$, that is, $$E_t(f)=\{\lambda\in i{\mathfrak{a}}^*\,:\,(\|\lambda\||{\mathbf{c}}(\lambda)|^{-2})^{nq/q'}|\mathcal{F}f(\lambda+\eta)|>t\}.$$ According to lemma \[lemma.NPP-2\](b), $E_t(f)$ is contained in the set $$\begin{split}
A_t(f)&:=\{\lambda\in i{\mathfrak{a}}^*\,:\, c_p(\|\lambda\||{\mathbf{c}}(\lambda)|^{-2})^{nq/q'}\|f\|_1>t\}\\
&=\{\lambda\in i{\mathfrak{a}}^*\,:\, \|\lambda\||{\mathbf{c}}(\lambda)|^{-2}>a_t\},\quad a_t=\Bigl(\frac{t}{c_p\|f\|_1}\Bigr)^{\frac{q'}{nq}},
\end{split}$$ We can now invoke the trick of Ray and Sarkar: Noting that $$\|\lambda\||{\mathbf{c}}(\lambda)|^{-2}\asymp \|\lambda\|^{2\vert\Sigma_0^+|+1}(1+\|\lambda\|)^{\beta-2|\Sigma_0^+|}=G(\|\lambda\|),$$ where 