machinery produces the desired inequality. In statement (i), one considers measure spaces $(S,dx)$ and $({\mathbb{R}}^\times,d\overline{\mu}(\lambda))$, where $d\overline{\mu}(\lambda)=|\lambda|^{-q}|{\mathbf{c}}(\lambda)|^{-2(1-q)}\,d\lambda$, and a sublinear operator $T$ defined for $f\in L^1(S)+L^q(S)$ by $Tf(\lambda)=\|\widetilde{f}(\lambda+i\gamma_q\rho,\cdot)\|_{L^q(N)}(|\lambda||{\mathbf{c}}(\lambda)|^{-2})^{q/q'}$. It can be verified that $T$ is of strong type $(q,q')$ (which follows from a Hausdorff–Young inequality and the Plancherel theorem) and weak type $(1,1)$ (which requires more work). An interpolation argument yields the conclusion in (i).

In (ii), it is convenient to consider measure spaces $(S,dx)$ and $({\mathbb{R}},|{\mathbf{c}}(\lambda)|^{-2}d\lambda)$ and a sublinear operator $T$ defined for $f\in L^1(S)+L^{p',1}(S)$ by $Tf(\lambda)=\|\widetilde{f}(\lambda+i\gamma_{q'}\rho,\cdot)\|_{L^{q'}(N)}$. While we shall not give the details of their interpolation argument, it also uses interpolation between Lorentz spaces, showing that for $q\leq<\infty$ and $1\leq s\leq\infty$, it holds that $\|Tf\|_{p,s}^*\leq C_{p,q}\|f\|^*_{p',s}$, cf. [@Ray-Sarkar-trans Eqn. (4.27)]. An important difference is that Ray and Sarkar use the function $J$ as Young function, which also dictates their definition of $\|f\|_{(p)}$. Let $u=\frac{p}{p-2}$ and $g(x)=f(x)J(x)^{1/u}$. Then $g\in L^p(S)$ and $\|g\|_p=\|f\|_{(p)}$. Moreover $m(\{x\in S\|:\| J(x)\leq t\})\leq Ct$ for all $t>0$, for some constant $C$, where $m$ is Haar measure on $S$ (so $J$ is indeed a Young function in our terminology). Consequently, $J(x)^{-1/u}\in L^{u,\infty}(S)$. It follows by Hölder’s inequality that $\|f\|^*_{p',p}\leq C_p\|g\|_p\|J^{-1/u}\|_{u,\infty}$, and therefore (taking $s=p$) $\|Tf\|_{p,p}^*\leq C_{p,q}\|g\|_p$, 