  In this article, we characterize the image of the Brylinski-Radon transform
in characteristic $p>0$ via Beilinson's theory of singular supports. We also
provide an alternate proof of Brylinski's results over $\mathbb{C}$, which also
works for sheaves with finite coefficients. Along the way, we also obtain a
microlocal criterion for the descent of perverse sheaves which could be of
independent interest.
