able to obtain a different version of the Hardy–Littlewood inequality for the Helgason–Fourier transform. They might have been motivated by the complex version of the Hausdorff–Young inequality, cf. . where the transform is extended holomorphically into a certain domain in the complex plane. At the same time Ray and Sarkar used slightly different weights in their interpolation argument, the result being the following theorem, cf. [@Ray-Sarkar-trans Theorem 4.11] (in their notation $S$ denotes a Damek–Ricci space, but the reader may replace it with a hyperbolic space).

\[thm.Ray-Sarkar\]

1.  Let $1<q\leq 2$ be fixed. Then for $f\in L^p(S)$, $1<p\leq q$, $$\Bigl(\int_{\mathbb{R}}\|\widetilde{f}(\lambda\pm i\gamma_q\rho,\cdot)\|_{L^q(N)}^r (|\lambda||{\mathbf{c}}(\lambda)|^{-2})^{r/p'-1}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\Bigr)^{1/r}\leq C_{\pm,p,q}\|f\|_p$$ where $\frac{1}{r}=1-\frac{q'-1}{p'}$.

2.  Let $2\leq q<\infty$ be fixed. Then for $f\in L^{(p)}(S)$ with $q\leq p<\infty$, $$\Bigl(\int_{\mathbb{R}}\|\widetilde{f}(\lambda\pm i\gamma_{q'}\rho,\cdot)\|^p_{L^{q'}(N)}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\Bigr)^{1/p}\leq C_{\pm,p,q}\|f\|_{(p)}.$$

Here $\|f\|_{(p)}=\Bigl(\int_S|f(x)|^pJ(x)^{p-2}\,dx\Bigr)^{1/p}$, where $J(x)=(\sinh\frac{r(x)}{2})^m(\sinh r(x))^l$ is essentially the Jacobian associated with polar coordinates in $S$. There are subtle technical issues pertaining to the proper domain of definition of the Helgason–Fourier transform that make the proof of theorem \[thm.Ray-Sarkar\] more involved than what might be expected, but the strategy of proof is still to use interpolation. Indeed the main task is again to identify two suitable measure spaces and a sublinear operator in such a way that the abstract interpolation 