simplify the proof of Eguchi and Kumahara by means of more refined interpolation techniques. These were later incorporated in [@Mohanty-II] where the authors established an analogue of for the Helgason–Fourier transform on a noncompact Riemannian symmetric space of rank one: It holds that $$\label{eqn.MRSS}
\int_{{\mathfrak{a}}^*} \|\widetilde{f}(\lambda,\cdot)\|_{L^1(K)}|\lambda|^{p-2}(1+|\lambda|)^{-(m_\gamma+m_{2\gamma})}|\mathbf{c}(\lambda)|^{-2}\,d\lambda\leq C\|f\|_p^p$$ for $1<p<2$. According to [@Mohanty-II Remark 4.6], their method also works for higher rank spaces. While we share this sentiment, it turns out to be slightly involved to fill in the necessary details. One may also object that the appearance of the average over $K$ is not natural. A different version was recently obtained in [@Ray-Sarkar-trans], to which we shall return later. A further drawback of is that for $p=2$ it does not resemble the Parseval identity, and section \[sec.HY-ineqs\] opens with the observation that the analogue of for the Heckman–Opdam transform, or even just the Jacobi transform in rank one, does not hold for arbitrary non-negative root multiplicities. We also wish to emphasize a quantitative difference between and : In the first inequality a weight is introduced on the function-side, whereas the second inequality incorporates a weight on the Fourier transform side. Theorem \[thm.1\] and theorem \[thm.2\] therefore resemble , whereas resembles 