Let $\psi$ be an smooth admissible Shearlet with compact support, cand let $\mathcal{M}$ be a bounded region in $\mathbb{R}^2$ and let $m= \chi_{\mathcal{M}}$ be the characteristic function of $\mathcal{M}$.

Now define $$ m_h = \mathcal{F}^{-1}[\hat{m}\chi_h], $$ where $h = \{(\xi_1,\xi_2)\in \mathbb{R}^2: |\xi_2|/|\xi_1|\leq 3/2\}$, and consider $$ M_{a,s}^h(x)=\int_{\mathbb{R}^2}\int_{\mathbb{R}^2}m_h(\eta)\psi_{a,s,t}(\eta)\psi_{a,s,t}(x)dtd\eta. $$ Here $a \in (0,1)$, $s\in (-2,2)$ and $t\in \mathbb{R}^2$ are the shearlet parameters. We define $$ \psi_{a,s,t}(x)=a^{-3/4}\psi(A_a^{-1}S_s^{-1}(x-t)), $$ where $A_a$ is a horizontal parabolic dilation matrix and $S_s$ is horizontal shearing matrix.

I have been able to show that $M_{a,s}^h$ is in all $L^p$ spaces and has zero integral. Obviously this means that $\widehat{M_{a,s}^h}$ is bounded and supported away from zero, but I'm not sure what else I am able to conclude about $\widehat{M_{a,s}^h}$. Specifically I am wondering what I can say about the decay of $\widehat{_{a,s}^h}$ as a function of $a$ or $x$.