f(x)\vert^q\|x\|^{n(q-2)}dx\Bigr)^{1/q}.$$

By duality and general properties of the Fourier transform, one has the following equivalent formulation: For every $p\in(1,2)$ there exists a positive constant $B_p$ independent of $f$ such that $$\label{HL-ineq2}
\Bigl(\int_{{\mathbb{R}}^n}|{\hat{f}}(\xi)|^p|\xi|^{n(p-2)}\,d\xi\Bigr)^{1/p}\leq B_p\Bigl(\int_{{\mathbb{R}}^n}|f(x)|^p\,dx\Bigr)^{1/p}.$$

An analogue of for the spherical transform on a Riemannian symmetric space $G/K$ was obtained by Eguchi and Kumahara in [@Eguchi-Kumahara Theorem 1, Section 5]:

\[thm.1\] Let $q\geq 2$. The spherical Fourier transform can be defined for $K$-invariant functions $f$ on $G/K$ with the property that $f\cdot\sigma^{n(1-2/q)}\Omega^{1-2/q}$ belongs to $L^q(K\setminus G/K)$, and there exists a positive constant $A_q$ that is independent of $f$ such that $$\label{ineq-HL-GK}
    \Bigl(\frac{1}{\vert W\vert}\int_{{\mathfrak{a}}^*}\vert\widetilde{f}(\lambda)\vert^q\,\vert\mathbf{c}(\lambda)\vert^{-2}\,d\lambda\Bigr)^{1/q} \leq A_q\Bigl(\int_G\vert f(x)\vert^q\sigma(x)^{n(q-2)}\Omega(x)^{q-2}\,dx\Bigr)^{1/q}$$ for all $f\in\mathcal{S}(K\setminus G/K)$.

Here $\sigma(x)=\left<X,X\right>^{1/2}$ where $\left<\cdot,\cdot\right>$ is the Cartan–Killing form and $G\ni x=k\exp X\in K\times\mathfrak{p}$, and $\Omega(\exp H)=c\prod_{\alpha\in\Sigma}\vert\sinh\alpha(H)\vert^{m(\alpha)}$, $H\in\mathfrak{a}$, the usual weight and $\mathcal{S}(K\setminus G/K)$ an $L^2$-based Schwartz space of $K$-invariant functions on $G/K$. An interpolation argument leads to an analogous statement for exponents below $2$:

\[thm.2\] Let $p\in(1,2]$ and $\frac{1}{p}+\frac{1}{q}=1$. Let $r\in[p,q]$ and set $\mu=\frac{1}{r}+\frac{1}{q}-1=\frac{1}{r}-\frac{1}{p}$. Then there exists a positive constant $B_r$ independent of $f$ such that $$\Bigl(\frac{1}{|W|}\int_{{\mathfrak{a}}^*}\vert\widetilde{f}(\lambda)\vert^q\vert\mathbf{c}(\lambda)\vert^{-2}\,d\lambda\Bigr)^{1/q} \leq B_r\Bigl(\int_G\vert f(x)\vert^r\sigma(x)^{-n\mu r}\Omega(x)^{-\mu r}\,dx\Bigr)^{1/r}$$ for all $f$ satisfying $f\cdot\sigma^{-n\mu}\Omega^{-\mu}\in L^r(K\setminus G/K)$.

It was remarked in the MathSciNet review by Michael Cowling that one could 