C_p\|f\|_p^p$$ for every $f\in L^p(A,d\mu)^W$.

Consider the measure spaces $(X,d\mu)=(A/W,d\mu)$ and $$(Y,d\overline{\nu})=\Bigl(i{\mathfrak{a}}^*, \frac{1}{\vert W\vert}\|\lambda\|^a(1+\|\lambda\|)^b|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\Bigr).$$ Let $Tf(\lambda)=\|\lambda\|^{k+n}\mathcal{F}f(\lambda)$, where $k$ is to be determined shortly. As in the rank one calculation that preceded the present theorem, the crux of the proof will be to verify that $T$ is strong type $(2,2)$ and weak type $(1,1)$ as an operator from $(X,d\mu)$ into $(Y,d\overline{\nu})$. As for the first property, it follows from Plancherel’s theorem that $$\|Tf\|_2^2 =\frac{1}{|W|}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2\|\lambda\|^{2(k+n)}\|\lambda\|^a(1+\|\lambda\|)^b|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda \simeq \|f\|_2^2$$ *provided* $\psi(\lambda)=\|\lambda\|^{2(k+n)}\|\lambda\|^a(1+\|\lambda\|)^b\asymp 1$. This holds whenever $2(k+n)+a+b\leq 0$ (the factor $\psi$ stays bounded for $\|\lambda\|\gg 1$) and at the same time $2(k+n)+a\geq 0$ (so that $\psi$ is bounded near $\lambda=0$).

For $t>0$ and $f\in L^1\cap L^2$, $\|f\|_1\neq 0$, consider the sets $E_t(f)=\{\lambda\,:\,\vert Tf(\lambda)\vert>t\}$ and $A_t(f)=\{\lambda\,:\,\|\lambda\|>(\frac{t}{C\|f\|_1})^{\frac{1}{k+n}}\}$, where $C$ is the constant coming from the estimate $\vert\mathcal{F}f(\lambda)|\leq C\|f\|_1$. By definition of $T$ it follows that $E_t(f)\subset A_t(f)$, whereby $$\begin{split}
|E_t(f)|& \leq |A_t(f)| =\frac{1}{|W|}\int_{A_t(f)} \|\lambda\|^a(1+\|\lambda\|)^b|{\mathbf{c}}(\lambda)\|^{-2}\,d\lambda\\ &\simeq \int_{A_t(f)} \|\lambda\|^{a+2\vert\Sigma_0^+\vert}(1+\|\lambda\|)^{b+\beta-2\vert\Sigma_0^+\vert}\,d\lambda\simeq \int_{A_t(f)}\|\lambda\|^{a+b+\beta}\,d\lambda \\ &= C'\int_{a_t}^\infty s^{a+b+\beta}s^{n-1}\,ds=C'\Bigl(\Bigl(\frac{t}{C\|f\|_1}\Bigr)^{\frac{1}{k+n}}\Bigr)^{a+b+\beta+n} = C''\frac{\|f\|_1}{t}
\end{split}$$ since $a+b+\beta+n=-(k+n)$ by construction[^1].

The extension to the case $p>2$ utilizes the stronger interpolation result theorem \[thm.lorentz-inter\] and is motivated by the rank one statement in [@Mohanty-II Theorem 4.5]. A similar argument leads to a generalization of [@Anker-besov Lemma §4.1] but we 