  In this article, we prove the $t$-exactness of a Brylinski-Radon
transformation taking values in sheaves on flag varieties. This implies several
weak Lefschetz type results for cohomology. In particular, we obtain de
Cataldo-Migliorini's P=Dec(F) and Beilinson's basic lemma, the latter was an
important ingredient in their proof of P=Dec(F). Our methods also allow the
sharpening of Esnault-Katz's cohomological divisibility theorem and estimates
for the Hodge level. Finally, we upgrade P=Dec(F) to an equivalence of functors
which is also valid over a base.
