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 $x$.



$f \in L^p(\mathbb{R}^2)$ for all $p \geq 1$, and $f$ has zero integral. What can we say about this function's fourier Transform?
Let $f$ be in $L^1(\mathbb R^n)$, then $\hat f$ belongs to $L^\infty(\mathbb R^n)$ and is (uniformly) continuous with $\lim\_{\vert \xi\vert\rightarrow +\infty} \hat f(\xi)=0$: this is the Riemann-Lebesgue Lemma.
If $f$ belongs to $L^p(\mathbb R^n)$, for some $p\in [1,2]$, then $\hat f$ belongs to $L^{p'}(\mathbb R^n)$: this is the Hausdorff-Young Theorem.


As a result your function $\hat f$ is continuous, belongs to $L^{q}(\mathbb R^n), 2\le q\le \infty$, goes to zero at infinity and vanishes at $\xi =0$. Although the map 
$L^1(\mathbb R^n)\ni f\mapsto \hat f\in C^0\_{(0)}(\mathbb R^n)$ is not onto
(the bottom (0) is for "going to 0 at infinity"), not much more can be said.


