where $$\psi(\lambda)=\begin{cases}\psi_1(\lambda)&\text{for }\|\lambda\|\leq 1\\ \psi_2(\lambda)&\text{for } \|\lambda\|>1\end{cases}$$ for suitable choices of $\psi_1,\psi_2$, but this leads to so much freedom that one should no longer speak of Hausdorff–Young inequalities. It would, however, allow one to treat the case $0\leq m_\alpha+m_{2\alpha}<1$ as well.

The next result may be seen as a weighted Hardy–Littlewood inequality, from which a natural analogue of the Hardy–Littlewood inequality in [@Mohanty-II] will follow. It is important to allow a certain freedom in the choice of weights, since it would otherwise be difficult to ‘guess’ the correct formulation in higher rank. The parameter constraints arise from having to be able to find an elementary proof of the required weak type $(1,1)$ estimate. Recall that $\beta=\sum_{\alpha\in\Sigma_0^+}(m_\alpha+m_{2\alpha})$, where $m:\Sigma\to[0,\infty)$ is a non-negative multiplicity function, and $m_\alpha:=m(\alpha)$.

\[prop.abstract-interpolate\] Assume $\beta+n>0$ and $1<p<2$, and consider the operator $T$ defined on $L^2(X,d\mu)$, $X=A/W$, by $Tf(\lambda)=\|\lambda\|^{k+n}\mathcal{F}f(\lambda)$, where $k\geq 0$. Moreover let $$(Y,d\overline{\nu})=\Bigl(i{\mathfrak{a}}^*, \frac{1}{\vert W\vert}\|\lambda\|^a(1+\|\lambda\|)^b|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\Bigr)$$ where the parameters $k,a,b$ satisfy the conditions

1.  $a+b\leq \frac{2}{3}(n-\beta)$

2.  $a+b+\beta+n=-(k+n)$

Then $T$ is of strong type $(2,2)$ and weak type $(1,1)$ as an operator from $L^2(X,d\mu)$ into $L^2(Y,d\overline{\nu})$, and therefore of strong type $(p,p)$. More precisely, there exists a positive constant $C_p$ independent of $f$ such that $$\frac{1}{|W|}\int_{i{\mathfrak{a}}^*}\vert\mathcal{F}f(\lambda)\vert^p \|\lambda\|^{(k+n)p+a}(1+\|\lambda\|)^b\,d\nu(\lambda)\leq 