$L^2(i{\mathfrak{a}}^*,|{\mathbf{c}}(\lambda)|^{-2}d\lambda)^W$. Moveover, for $f\in C_c^\infty({\mathfrak{a}})^W$, $$f(x)=\int_{i{\mathfrak{a}}^*}\mathcal{F}f(\lambda)\varphi_{-\lambda}(x)|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda$$ for all $x\in{\mathfrak{a}}$.

For $p\in(0,2]$, set $\epsilon_p=\frac{2}{p}-1$. Let $C(\epsilon_p)$ be the convex hull in ${\mathfrak{a}}^*$ of the set $\{\epsilon_p w\rho\,:\,w\in W\}$, and let ${\mathfrak{a}}^*_{\epsilon_p}=C(\epsilon_p\rho)+i{\mathfrak{a}}^*$.

The following two versions of the Hausdorff–Young theorem for the Heckman–Opdam transform were recently established by Narayanan, Pasquale, and Pusti, cf. [@Narayanan-Pasquale-Pusti Lemma 5.2, Lemma 5.3]. The first proof involves Riesz–Thorin interpolation, whereas the second uses interpolation with an analytic family of operators.

Let $f\in L^p(A,d\mu)^W$. Then the following properties hold.

1.  The hypergeometric transform $\mathcal{F}f(\lambda)$ is well defined for all $\lambda$ in the interior of ${\mathfrak{a}}^*_{\epsilon_p}$ and defines a holomorphic function.

2.  Let $p,q$ be so that $1<p<2$ and $\frac{1}{p}+\frac{1}{q}=1$. Then there exists a positive constant $c_p$ independent of $f$ so that $$\label{ineq-HY-root-1}
    \|\mathcal{F}f\|_q = \Bigl(\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^q\,d\nu(\lambda)\Bigr)^{1/q}\leq c_p\|f\|_p.$$

A more precise formulation is given as follows. Let $f\in L^1({\mathfrak{a}},d\mu)^W\cap L^2({\mathfrak{a}},d\mu)^W$. If $1\leq p\leq 2$, $q=\frac{p}{p-1}$, then $\|\mathcal{F}f\|_{q}\leq c_p\|f\|_p$. Since $L^p$ is dense in $L^1\cap L^2$ for $1\leq p\leq 2$, the Heckman–Opdam transform $\mathscr{F}_pf$ can be defined uniquely for all $f\in L^p({\mathfrak{a}},d\mu)^W$, $1\leq p\leq 2$, so that $\mathscr{F}_p:L^p({\mathfrak{a}},d\mu)^W\to L^q(i{\mathfrak{a}}^*,d\nu)^W$ is a linear contraction with $\mathscr{F}_pf=\mathcal{F}f$ for all $f\in L^1({\mathfrak{a}},d\mu)^W\cap L^2({\mathfrak{a}},d\mu)^W$. It is known from [@Narayanan-Pasquale-Pusti Theorem 5.4] that $\mathscr{F}_p$ is injective 