  We prove a continuity result for the shearlet transform when restricted to
the space of smooth and rapidly decreasing functions with all vanishing
moments. We define the dual shearlet transform, called here the shearlet
synthesis operator, and we prove its continuity on the space of smooth and
rapidly decreasing functions over
$\mathbb{R}^2\times\mathbb{R}\times\mathbb{R}^\times$. Then, we use these
continuity results to extend the shearlet transform to the space of Lizorkin
distributions, and we prove its consistency with the classical definition for
test functions.
