on $L^p({\mathfrak{a}},d\mu)^W$ whenever $p\in[1,2]$.

\[lemma.NPP-2\] Let $f\in L^p({\mathfrak{a}},d\mu)^W$ for some $p\in(1,2)$ and let $\eta$ be in the interior of $C(\epsilon_p\rho)$. Then the following properties hold:

1.  Assume $\frac{1}{p}+\frac{1}{q}=1$. There exists a positive constant $C_{p,\eta}$ such that for all $f\in L^p({\mathfrak{a}},d\mu)^W$, $$\label{ineq-HY-root-2}
    \Bigl(\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda+\eta)|^q|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\Bigr)^{1/q}\leq C_{p,\eta}\|f\|_p.$$

2.  It holds that $\sup_{\lambda\in i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda+\eta)|\leq C_{p,\eta}\|f\|_p$ and $$\lim_{\lambda\in{\mathfrak{a}}^*_{\epsilon_p},|\Im\lambda|\to\infty} |\mathcal{F}f(\lambda)|=0.$$

An important ingredient in the proof of both results is a characterization of the set of spectral parameters $\lambda$ for which $\varphi_\lambda$ is bounded. This description was obtained in [@Narayanan-Pasquale-Pusti Theorem 4.2]: $\varphi_\lambda$ is bounded if and only if $\lambda\in C(\rho)+i{\mathfrak{a}}^*$, in which case $|\varphi_\lambda(x)|\leq 1$ for every $x\in{\mathfrak{a}}$. Note that $C(\epsilon_p\rho)\subset C(\rho)$ for $p\geq 2$. Also note that $C(\epsilon_2\rho)=\{0\}$.

\[lemma.p1p2\]

1.  If $f$ belongs to $(L^{p_1}\cap L^{p_2})({\mathfrak{a}},d\mu)^W$ for some $p_1,p_2\in[1,2]$, then $\mathscr{F}_{p_1}f=\mathscr{F}_{p_2}f$ $\nu$-almost everywhere on $i{\mathfrak{a}}^*$.

2.  If $h$ belongs to $(L^{q_1}\cap L^{q_2})(i{\mathfrak{a}}^*,d\nu)^W$ for some $q_1,q_2\in[1,2]$, then $\mathscr{I}_{q_1}h=\mathscr{I}_{q_2}h$ $\mu$-almost everywhere on ${\mathfrak{a}}$.

<!-- -->

1.  Choose a sequence $\{g_n\}_{n=1}^\infty$ of simple $W$-invariant functions on ${\mathfrak{a}}$ such that $$\lim_{n\to\infty}\|f-g_n\|_{p_1}=\lim_{n\to\infty}\|f-g_n\|_{p_2}=0.$$ Each function $\mathcal{F}g_n$ belongs to $(L^{p_1^\prime}\cap L^{p_2^\prime})(i{\mathfrak{a}}^*,d\nu)^W$ by the Hausdorff–Young inequality , and $$\lim_{n\to\infty}\|\mathscr{F}_{p_1}f-\mathcal{F}g_n\|_{p_1^\prime}=\lim_{n\to\infty}\|\mathscr{F}_{p_2}f\mathcal{F}g_n\|_{p_2^\prime}=0.$$ One can therefore extract subsequences $\{\mathcal{F}g_{n_k}\}_{k=1}^\infty$ and $\{\mathcal{F}g_{n_l}\}_{l=1}^\infty$ of $\{\mathcal{F}g_n\}_{n=1}^\infty$ such that $\mathcal{F}g_{n_k}\to\mathscr{F}_{p_1}f$ and $\mathcal{F}g_{n_l}\to\mathscr{F}_{p_2}f$ $\nu$-almost everywhere on $i{\mathfrak{a}}^*$, from which it follows that 