type $(2,2)$ as an operator from $L^2(A,d\mu)^W$ into $L^2(i{\mathfrak{a}}^*,d\overline{\nu})^W=L^2(i{\mathfrak{a}}^*,|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}d\nu(\lambda))^W$. This is no longer true when $m_\alpha+m_{2\alpha}<1$, in which case one would have to employ a different type of weight and/or modify the measure $d\overline{\nu}$.

Note that $|\mathcal{F}f(\lambda)|\leq C\|f\|_1$ for all $\lambda\in i{\mathfrak{a}}^*$ and $f\in L^1(A,d\mu)$.. For $t>0$ and $0\neq f\in L^1(A,d\mu)^W$, define $$E_t(f)=\{\lambda\,:\,|Tf(\lambda)|>t\},\quad 
A_t(f)=\Bigl\{\lambda\,:\,|\lambda|>\Bigl(\frac{t}{C\|f\|_1}\Bigr)^{1/2}\Bigr\},\quad\text{and}\quad
a_t =\Bigl(\frac{t}{C\|f\|_1}\Bigr)^{1/2}.$$ It follows from the definition of $T$ that $E_t(f)\subset A_t(f)$ for all $t>0$, hence $|E_t(f)|\leq|A_t(f)|$, where $$\begin{split}
|A_t(f)| &= \frac{1}{|W|}\int_{A_t(f)} (1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\\
&\asymp \frac{1}{|W|}\int_{A_t(f)} (1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}|\lambda|^2(1+|\lambda|)^{m_\alpha+m_{2\alpha}-2}\,d\lambda\\
&=\frac{1}{|W|}\int_{A_t(f)}|\lambda|^{-2}(1+|\lambda|)^{-1}\,d\lambda=\frac{1}{|W|}\int_{a_t}^\infty |\lambda|^{-2}(1+|\lambda|)^{-1}\,d\lambda\\
&\approx \frac{1}{|W|}\int_{a_t}^\infty\frac{d\lambda}{\lambda^3} = Ca_t^{-2}=C'\frac{\|f\|_1}{t}
\end{split}$$ This shows that $T$ is also of weak type $(1,1)$ as an operator from $L^2(A,d\mu)^W$ into $L^2(i{\mathfrak{a}}^*,d\overline{\nu})^W$. It follows from the Marcinkiewicz interpolation theorem that $T$ is of strong type $(p,p)$, for $p\in(1,2)$, that is $$\label{eqn.inter-ineq}
\frac{1}{|W|}\int_{i{\mathfrak{a}}^*}|Tf(\lambda)|^p(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}\,d\nu(\lambda)\leq C\|f\|_p^p.$$ Since $1+|\lambda|\geq|\lambda|^{2-p}$ for all $\lambda$, it holds that $(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}\geq|\lambda|^{2-p}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})}$, which allows us to rewrite as $$\frac{1}{|W|}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^p|\lambda|^{p-2}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})}\,d\nu(\lambda)\leq C\|f\|_p^p.$$

Consider the more natural weighted measure $d\widetilde{\nu}(\lambda)=|W|^{-1}|\lambda|^{-4}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda$. The operator $T$ is then of strong type $(2,2)$ as an operator from $L^2(A,d\mu)^W$ into $L^2(i{\mathfrak{a}}^*,d\widetilde{\nu})^W$ *for all* $m_\alpha,m_{2\alpha}$ but it is no clear if $T$ is of weak type $(1,1)$. A good choice of weights is therefore essential.

It is also possible to consider weighted measures of the form $d{\hat{\mu}}(\lambda)=\psi(\lambda)|{\mathbf{c}}(\lambda)|^{-2}d\lambda$ 